Exponential and

Logarithmic Relations
Chapter Overview and Pacing

PACING (days)
Regular Block
LESSON OBJECTIVES Basic/ Basic/
Average Advanced Average Advanced
Exponential Functions (pp. 522–530) 1 1 0.5 0.5
Preview: Investigating Exponential Functions
• Graph exponential functions.
• Solve exponential equations and inequalities.
Logarithms and Logarithmic Functions (pp. 531–540) 2 2 1 1
• Evaluate logarithmic expressions. (with 10-2
• Solve logarithmic equations and inequalities. Follow-Up)
Follow-Up: Modeling Real-World Data: Curve Fitting
Properties of Logarithms (pp. 541–546) 1 1 0.5 0.5
• Simplify and evaluate expressions using the properties of logarithms.
• Solve logarithmic equations using the properties of logarithms.
Common Logarithms (pp. 547–553) 1 1 0.5 0.5
• Solve exponential equations and inequalities using common logarithms.
• Evaluate logarithmic expressions using the Change of Base Formula.
Follow-Up: Solving Exponential and Logarithmic Equations and Inequalities
Base e and Natural Logarithms (pp. 554–559) 2 2 1 1
• Evaluate expressions involving the natural base and natural logarithms. (with 10-4 (with 10-4 (with 10-4 (with 10-4
• Solve exponential equations and inequalities using natural logarithms. Follow-Up) Follow-Up) Follow-Up) Follow-Up)
Exponential Growth and Decay (pp. 560–565) 1 1 0.5 0.5
• Use logarithms to solve problems involving exponential decay.
• Use logarithms to solve problems involving exponential growth.
Study Guide and Practice Test (pp. 566–571) 1 1 0.5 0.5
Standardized Test Practice (pp. 572–573)
Chapter Assessment 1 1 0.5 0.5
TOTAL 10 10 5 5

Pacing suggestions for the entire year can be found on pages T20–T21.

520A Chapter 10 Exponential and Logarithmic Relations

 Timesaving Tools
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All-In-One Planner
and Resource Center
Chapter Resource Manager See pages T12–T13.

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Materials
573–574 575–576 577 578 GCS 45 10-1 10-1 (Preview: paper, scissors, grid paper,
calculator)
graphing calculator, grid paper, string

579–580 581–582 583 584 623 SC 19 10-2 10-2 posterboard

(Follow-Up: graphing calculator,
grid paper)

585–586 587–588 589 590 623, 625 10-3 10-3

591–592 593–594 595 596 10-4 10-4 (Follow-Up: graphing calculator)

597–598 599–600 601 602 624 SM 127–132 10-5 10-5 19 plastic coins, paper currency

603–604 605–606 607 608 624 GCS 46, 10-6 10-6

SC 20

609–622,
626–628

*Key to Abbreviations: GCS  Graphing Calculator and Speadsheet Masters,

SC  School-to-Career Masters,
SM  Science and Mathematics Lab Manual

Chapter 10 Exponential and Logarithmic Relations 520B

 Mathematical Connections
and Background
Continuity of Instruction Exponential Functions
Examine the list of characteristics for an exponen-
tial function on page 524. The first characteristic states
that an exponential function is continuous and one-to-one.
The term continuous means that the function can be traced
Prior Knowledge without lifting your pencil. The term one-to-one means that
Students have worked with exponents in a horizontal line passing through the graph will intersect
many situations, including performing calcula- no more than one point on the graph. This characteristic
is important for the development of the logarithmic func-
tions, manipulating expressions, and applying
tion in Lesson 10-2, since only one-to-one functions can
properties. They have explored properties of have inverses. The second characteristic listed is that the
inverses for operations and for functions, and domain of the function is the set of all real numbers. This
they have solved many kinds of equations and property is important because it means that 35 has
inequalities. meaning, since 5 is a real number and part of the
domain of y  3x. The third and fourth characteristics of
an exponential function are related. The x-axis is a horizon-
tal asymptote of the graph of an exponential function. This
means that the graph of this function approaches the hor-
izontal line x  0, getting closer and closer to this line but
This Chapter never crossing it. This restricts the graph of an exponen-
tial function to either Quadrants I and II, when a is posi-
Students are introduced to the term logarithm tive, or to Quadrants III and IV, when a is negative. In
to solve for a variable that appears as an ex- terms of the range of the function, this means that when
ponent. They explore the relationship between a is positive, all y values of the function will be positive,
exponents and logarithms, and they use loga- and when a is negative, all y values of the function will
rithms with two special bases, base 10 or com- be negative. These two properties will also be important
mon logarithms, and base e or natural loga- when considering the inverse of the exponential function.
The last two properties are useful for graphing and
rithms. They apply the Change of Base Formula writing exponential functions.
to rewrite a logarithm using a different base,
and they apply appropriate formulas to solve Logarithms and
problems involving exponential growth
Logarithmic Functions
and exponential decay.
In the equation y  logb x, y is referred to as the
logarithm, b is the base, and x is sometimes referred to as
the argument. The definition of a logarithm given on
page 532 indicates that a logarithm is an exponent.
When solving logarithmic equations and
Future Connections inequalities, it is important to remember that a defining
characteristic of a logarithmic function is that its domain
Students will continue to look at properties is the set of all positive numbers. This means that the loga-
of and relationships between exponents and rithm of 0 or of a negative number for any base is unde-
logarithms. They will apply formulas for fined. It is very important to check possible solutions to
exponential growth and exponential decay logarithmic equations in the original equation, to be sure
in science courses and in consumer situations. that they would not result in taking the logarithm of 0 or
a negative number. For logarithmic inequalities, this fact
The natural-base exponential function will will exclude not just one value from the solution set, but
have an important role in precalculus and a range of values. In Example 8 on page 534, since the
calculus topics. original inequality asks for the values log10 (3x  4) and
log10 (x  6), we must solve two inequalities, 3x  4  0
and x  6  0, to find what values must be excluded from

520C Chapter 10 Exponential and Logarithmic Relations

the solution set we found using the Property of 645,000  6.45 105 Scientific notation
Inequality for Logarithmic Functions. Excluding the log 645,000  log (6.45 105) Property of Equality for
4
values such that x   and x  6, the solution set is Log Functions
3
all x such that the following three inequalities are all  log 6.45 + log 105 Product Property
4
satisfied: x  , x  6, and x  5. To simplify this  log 6.45 + 5 Inverse Property of
3 Exponents and Logs
compound inequality, sketch all three inequalities, as
 0.8096 + 5 log 6.45  0.8096
shown below, and find where all three intersect.
 5.8096 Simplify.
x 4–
3 log 645,000  5.8096
6 4 2 0 2 4 6 ↑ ↑
x  6 characteristic mantissa
6 4 2 0 2 4 6
x5 Base e and Natural
6 4 2 0 2 4 6
Logarithms
6 4 2 0 2 4 6 Exponentiation, which is the inverse operation
of taking a logarithm, is sometimes referred to as
The final number line shows that the solution set is finding the antilogarithm. That is, if log x  a then
4
the compound inequality   x  5. x  antilog a. Since antilogarithms mean the same
3
operation as exponentiation, it follows that to find the
antilogarithm of a common logarithm, you would use
Properties of Logarithms
2nd [10x ] on a graphing calculator. To find the
The word logarithm is actually a contraction of antilogarithm of a natural logarithm, antiln a, you
o
“lgical arit
h
metic.” Logarithms were invented to make
would use 2nd [e x ].
computation easier. Using logarithms, multiplication
changes to addition, according to the Product Property
of Logarithms, and division changes to subtraction, Exponential Growth and Decay
according to the Quotient Property of Logarithms. This It is important to note that the variable r in the
is illustrated in Examples 1, 2, and 4 of Lesson 10-3. In exponential decay formula y  a(1  r)t and the vari-
these examples, students are given the approximate able k in the alternate exponential decay formula
value of specific logarithms. Before the invention of the y  aekt are not equivalent. In a problem where a
scientific calculator, these values took a good deal of decay factor is given or asked for, the formula
time to compute. Rather than use the same arduous y  a(1  r)t should be used and not the formula
process to compute each and every logarithm one y  aekt. The same is true of the exponential growth
encountered, the properties of logarithms allowed the formulas y  a(1  r)t and y  ae kt.
use of a relative few logarithmic values to compute
others.

Common Logarithms
Before the invention of the scientific calculator,
the appendices of algebra texts contained extensive
tables of common logarithms of numbers. In order to www.algebra2.com/key_concepts
read these tables, you had to understand the parts of a
logarithm. Every logarithm has two parts, the character- Additional mathematical information and teaching notes
istic and the mantissa. A mantissa is the logarithm of a are available in Glencoe’s Algebra 2 Key Concepts:
number between 1 and 10. When the original number Mathematical Background and Teaching Notes,
is expressed in scientific notation, the characteristic is which is available at www.algebra2.com/key_concepts.
the power of 10. The lessons appropriate for this chapter are as follows.
• Exponential Functions (Lesson 33)
• Growth and Decay (Lesson 34)

Chapter 10 Exponential and Logarithmic Relations 520D

 and Assessment

Type Student Edition Teacher Resources Technology/Internet

Ongoing Prerequisite Skills, pp. 521, 530, 5-Minute Check Transparencies Alge2PASS: Tutorial Plus
INTERVENTION

538, 546, 551, 559 Quizzes, CRM pp. 623–624 www.algebra2.com/self_check_quiz

Practice Quiz 1, p. 538 Mid-Chapter Test, CRM p. 625 www.algebra2.com/extra_examples
Practice Quiz 2, p. 559 Study Guide and Intervention, CRM pp. 573–574,
579–580, 585–586, 591–592, 597–598, 603–604
Mixed pp. 531, 538, 546, 551, 559, 565 Cumulative Review, CRM p. 626
Review
Error Find the Error, pp. 535, 544, 557 Find the Error, TWE pp. 535, 544, 557
Analysis Common Misconceptions, p. 523 Unlocking Misconceptions, TWE pp. 542, 548
Tips for New Teachers, TWE p. 534
Standardized pp. 530, 537, 538, 546, 551, TWE p. 562 Standardized Test Practice
Test Practice 559, 562, 563, 564, 572–573 Standardized Test Practice, CRM pp. 627–628 CD-ROM
www.algebra2.com/
standardized_test
Open-Ended Writing in Math, pp. 530, 537, Modeling: TWE pp. 530, 565
Assessment 546, 551, 559, 564 Speaking: TWE pp. 546, 559
Open Ended, pp. 527, 535, 544, Writing: TWE pp. 538, 551
ASSESSMENT

549, 557, 563 Open-Ended Assessment, CRM p. 621

Chapter Study Guide, pp. 566–570 Multiple-Choice Tests (Forms 1, 2A, 2B), TestCheck and Worksheet Builder
Assessment Practice Test, p. 571 CRM pp. 609–614 (see below)
Free-Response Tests (Forms 2C, 2D, 3), MindJogger Videoquizzes
CRM pp. 615–620 www.algebra2.com/
Vocabulary Test/Review, CRM p. 622 vocabulary_review
www.algebra2.com/chapter_test

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Additional Intervention Resources TestCheck and Worksheet Builder

The Princeton Review’s Cracking the SAT & PSAT This networkable software has three modules for intervention
The Princeton Review’s Cracking the ACT and assessment flexibility:
• Worksheet Builder to make worksheet and tests
ALEKS • Student Module to take tests on screen (optional)
• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and
End-of-Course tests.

520E Chapter 10 Exponential and Logarithmic Relations

 Reading and Writing
in Mathematics
Intervention Technology Glencoe Algebra 2 provides numerous opportunities to
Alge2PASS: Tutorial Plus CD-ROM offers a incorporate reading and writing into the mathematics
complete, self-paced algebra curriculum. classroom.

Algebra 2 Alge2PASS Lesson Student Edition

Lesson
• Foldables Study Organizer, p. 521
10-5 19 Exponential and Logarithmic Functions • Concept Check questions require students to verbalize
and write about what they have learned in the lesson.
ALEKS is an online mathematics learning system that (pp. 527, 535, 544, 549, 557, 563, 566)
adapts assessment and tutoring to the student’s needs. • Writing in Math questions in every lesson, pp. 530, 537,
Subscribe at www.k12aleks.com. 546, 551, 559, 564
• WebQuest, pp. 529, 565

Intervention at Home Teacher Wraparound Edition

• Foldables Study Organizer, pp. 521, 566
Log on for student study help. • Study Notebook suggestions, pp. 522, 527, 535, 544,
• For each lesson in the Student Edition, there are Extra 549, 557, 563
Examples and Self-Check Quizzes. • Modeling activities, pp. 530, 565
www.algebra2.com/extra_examples • Speaking activities, pp. 546, 559
www.algebra2.com/self_check_quiz • Writing activities, pp. 538, 551
• For chapter review, there is vocabulary review, test • ELL Resources, pp. 520, 529, 537, 545, 550, 558,
practice, and standardized test practice. 564, 566
www.algebra2.com/vocabulary_review
www.algebra2.com/chapter_test Additional Resources
www.algebra2.com/standardized_test
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
they progress through the chapter. (Chapter 10 Resource
For more information on Intervention and Masters, pp. vii-viii)
Assessment, see pp. T8–T11. • Reading to Learn Mathematics master for each lesson
(Chapter 10 Resource Masters, pp. 577, 583, 589, 595,
601, 607)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom
• WebQuest and Project Resources

For more information on Reading and Writing in

Mathematics, see pp. T6–T7.
Chapter 10 Exponential and Logarithmic Relations 520F
 Notes Exponential and
Logarithmic
Have students read over the list
Relations
of objectives and make a list of
any words with which they are
not familiar. • Lessons 10-1 through 10-3 Simplify exponential
Key Vocabulary
and logarithmic expressions. • exponential growth (p. 524)
• Lessons 10-1, 10-4, and 10-5 Solve exponential • exponential decay (p. 524)
equations and inequalities. • logarithm (p. 531)
• Lessons 10-2 and 10-3 Solve logarithmic • common logarithm (p. 547)
equations and inequalities.
• natural logarithm (p. 554)
Point out to students that this is • Lesson 10-6 Solve problems involving
only one of many reasons why exponential growth and decay.
each objective is important.
Others are provided in the
introduction to each lesson.
Exponential functions are often used to model problems involving
growth and decay. Logarithms can also be used to solve such
problems. You will learn how a declining farm
population can be modeled by an exponential
function in Lesson 10-1.

NCTM Local
Lesson Standards Objectives
10-1 1, 2, 3, 6, 7, 8,
Preview 10
10-1 1, 2, 3, 4, 6, 8,
9, 10
10-2 1, 2, 3, 4, 6, 7,
8, 9
10-2 1, 2, 3, 5, 6, 8,
Follow-Up 10
10-3 1, 2, 4, 6, 7, 8,
9
10-4 1, 2, 4, 6, 8, 9
10-4 1, 2, 3
520 Chapter 10 Exponential and Logarithmic Relations
Follow-Up
10-5 1, 2, 3, 4, 6, 7,
8, 9
10-6 1, 2, 4, 6, 8, 9
Vocabulary Builder ELL

The Key Vocabulary list introduces students to some of the main vocabulary terms
Key to NCTM Standards: included in this chapter. For a more thorough vocabulary list with pronunciations of
1=Number & Operations, 2=Algebra, new words, give students the Vocabulary Builder worksheets found on pages vii and
3=Geometry, 4=Measurement, viii of the Chapter 10 Resource Masters. Encourage them to complete the definition
5=Data Analysis & Probability, 6=Problem of each term as they progress through the chapter. You may suggest that they add
Solving, 7=Reasoning & Proof, these sheets to their study notebooks for future reference when studying for the
8=Communication, 9=Connections, Chapter 10 test.
10=Representation
520 Chapter 10 Exponential and Logarithmic Relations
Prerequisite Skills To be successful in this chapter, you’ll need to master This section provides a review of
these skills and be able to apply them in problem-solving situations. Review the basic concepts needed before
these skills before beginning Chapter 10.
beginning Chapter 10. Page
Lessons 10-1 through 10-3 Multiply and Divide Monomials
references are included for
additional student help.
Simplify. Assume that no variable equals 0. (For review, see Lesson 5-1.)
36x7y4z3 12x3 4ab2 2 a2
4. 
64b3c  256b2c2
1. x5  x  x6 x12 2. (3ab4c2)3 27a 3b12c 6 3. 4   Prerequisite Skills in the Getting
21x y9z4 7y5z
Ready for the Next Lesson section
Lessons 10-2 and 10-3 Solve Inequalities at the end of each exercise set
Solve each inequality. (For review, see Lesson 1-5) review a skill needed in the next
5. a  4  10 6. 5n  15 7. 3y  2  4 8. 15  x 9 lesson.
a  14 n  3 y  2 x6
Lessons 10-2 and 10-3 Inverse Functions
Find the inverse of each function. Then graph the function and its inverse. 9. f 1(x)  1x For Prerequisite
(For review, see Lesson 7-8.) 912. See pp. 573A–573D for graphs. 2 Lesson Skill
x4
9. f(x)
2x 10. f(x)
3x  2 11. f(x)
x  1 12. f(x)
 10-2 Composition of Functions
3
f 1(x)  x2 f 1(x)  x  1 f 1(x)  3x  4 (p. 530)

Lessons 10-2 and 10-3 3 Composition of Functions
10-3 Multiplying and Dividing
Find g[h(x)] and h[g(x)]. (For review, see Lesson 7-7.) Monomials (p. 538)
13. h(x)
3x  4 g[h(x)]  3x  2 14. h(x)
2x  7 g[h(x)]  10x  35
10-4 Solving Logarithmic Equations
g(x)
x  2 h[g(x)]  3x  2 g(x)
5x h[g(x)]  10x  7 and Inequalities (p. 546)
15. h(x)
x  4 g[h(x)]  x2  8x  16 16. h(x)
4x  1 g[h(x)]  8x  5 10-5 Logarithmic Equations (p. 551)
g(x)
x2 h[g(x)]  x2  4 g(x)
2x  3 h[g(x)]  8x  11
10-6 Exponential Equations and
Inequalities (p. 559)

Make this Foldable to record information about exponential

and logarithmic relations. Begin with four sheets of grid paper.

Fold and Cut Fold and Label

First Sheets Second Sheets

Insert first sheets
through second sheets
and align folds. Label
pages with lesson
numbers.

Fold in half along

the width. On the first two
sheets, cut along the fold at
the ends. On the second two
sheets, cut in the center of
the fold as shown.

Reading and Writing As you read and study the chapter, fill the journal with notes,
diagrams, and examples for each lesson.

Chapter 10 Exponential and Logarithmic Relations 521

TM

Organization of Data and Journal Writing After students

make their Foldable journals, have them label two pages for each
lesson in Chapter 10. Writers’ journals can be used by students to
For more information record the direction and progress of learning, to describe positive
about Foldables, see and negative experiences during learning, to write about personal
Teaching Mathematics associations and experiences called to mind during learning, and
with Foldables. to list examples of ways in which new knowledge has or will be
used in their daily life, as well as take notes, record key concepts,
and write examples.

Chapter 10 Exponential and Logarithmic Relations 521

Algebra
Activity A Preview of Lesson 10-1

A Preview of Lesson 10-1

Getting Started Investigating Exponential Functions

Collect the Data
Objective Use paper stacking to Step 1 Cut a sheet of notebook paper in half.
explore an exponential function.
Step 2 Stack the two halves, one on top of the other.
Materials Step 3 Make a table like the one below and record
notebook paper the number of sheets of paper you have in
scissors the stack after one cut.
grid paper
Number of Cuts Number of Sheets
0 1

Teach 1 2
2 4
• You may wish to do the
example as a demonstration
while students complete the Step 4 Cut the two stacked sheets in half, placing the resulting pieces in a single
table on the chalkboard. stack. Record the number of sheets of paper in the new stack after 2 cuts.
• Students may recognize that the Step 5 Continue cutting the stack in half, each time putting the resulting piles in a
y value is doubled for each suc- single stack and recording the number of sheets in the stack. Stop when the
resulting stack is too thick to cut.
cessive cut, but they may have
to be led to realizing that this 1. (0, 1),
Analyze the Data (1, 2), (2, 4),
can be written in the form 2x.
1. Write a list of ordered pairs (x, y), where x is the number of cuts and y is the (3, 8),
• Show students how to connect number of sheets in the stack. Notice that the list starts with the ordered pair (4, 16), …
the points with a smooth curve, (0, 1), which represents the single sheet of paper before any cuts were made. 2. (5, 32), (6,
rather than connecting each 2. Continue the list, beyond the point where you stopped cutting, until you reach 64), (7, 128);
the ordered pair for 7 cuts. Explain how you calculated the last y values for your The y value is
pair of points with a straight list, after you had stopped cutting. found by
line. 3. Plot the ordered pairs in your list on a coordinate grid. Be sure to choose a scale raising 2 to
for the y-axis so that you can plot all of the points. See pp. 573A–573D. the number of
4. Describe the pattern of the points you have plotted. Do they lie on a straight line?
cuts.
Assess The points do not lie in a straight line. The slope increases as the x values increase.
Make a Conjecture
Have students work in small 5. Write a function that expresses y as a function of x. y  2x
groups for Exercises 1–9. 6. Use a calculator to evaluate the function you wrote in Exercise 5 for x
8 and
Observe students’ work to x
9. Does it give the correct number of sheets in the stack after 8 and 9 cuts? 256, 512; yes
determine if they are able to 7. Notebook paper usually stacks about 500 sheets to the inch. How thick would
your stack of paper be if you had been able to make 9 cuts? about 1 in.
write the function in Exercise 5.
8. Suppose each cut takes about 5 seconds. If you had been able to keep cutting,
Students should conclude after you would have made 36 cuts in three minutes. At 500 sheets to the inch, make
Exercise 9 that exponential a conjecture as to how thick you think the stack would be after 36 cuts. Sample answer:
functions can increase faster than 9. Use your function from Exercise 5 to calculate the thickness of your stack after 1 million ft
seems reasonable. 36 cuts. Write your answer in miles. 2169 mi

522 Investigating Slope-Intercept Form

522 Chapter 10 Exponential and Logarithmic Relations

Resource Manager
Teaching Algebra with Glencoe Mathematics Classroom
Study Notebook Manipulatives Manipulative Kit
• p. 1 (grid paper) • scissors
You may wish to have students • p. 275 (student recording sheet) • coordinate grid stamp
summarize this activity and what
they learned from it.

522 Chapter 10 Exponential and Logarithmic Relations

 Exponential Functions Lesson
Notes

• Graph exponential functions.

• Solve exponential equations and inequalities.
1 Focus
Vocabulary does an exponential function describe tournament play?
• exponential function The NCAA women’s 5-Minute Check
• exponential growth basketball tournament 2001 NCAA Women’s Tournament Transparency 10-1 Use as
• exponential decay begins with 64 teams and a quiz or review of Chapter 9.
• exponential equation consists of 6 rounds of SW Mo. State Connecticut
• exponential inequality play. The winners of the SW Mathematical Background notes
first round play against WEST Missouri Connecticut EAST
State are available for this lesson on
each other in the second
round. The winners then Washington Purdue La. Tech p. 520C.
move from the Sweet Sixteen Notre
to the Elite Eight to the Dame Building on Prior
Final Four and finally to
Xavier Notre Dame Knowledge
the Championship Game. Notre Dame

The number of teams y MIDEAST Purdue Notre Dame MIDWEST Ask students where they have
that compete in a heard the term exponential before
tournament of Purdue Vanderbilt and what they think it might
x rounds is y
2x. mean. Students may have heard
terms like exponential growth on a
television news program and they
Study Tip EXPONENTIAL FUNCTIONS In an exponential function like y
2x, the base might think that exponential
is a constant, and the exponent is a variable. Let’s examine the graph of y
2x.
Common means “enormous.” Use students’
Misconception answers to introduce the concept
Be sure not to confuse Example 1 Graph an Exponential Function of exponential functions.
polynomial functions and
exponential functions. Sketch the graph of y  2x. Then state the function’s domain and range.
While y
x2 and y
2x Make a table of values. Connect the points to sketch a smooth curve. does an exponential
each have an exponent, function describe
y
x2 is a polynomial y
function and y
2x is an x y  2x 8
y  2x
tournament play?
exponential function.
3 23
1

 7
Ask students:
8
1 2 7  6.3 • How many winners are there
2 22
 6
4 in the first round of the
1
1 21
 5 Notice that tournament? 32
2 the domain
0 20
1 4 of y  2x • After each round, how has the
1 As the value of includes number of teams changed? The
1  3
 2 
2
2 x decreases, irrational
2
the value of y numbers number of teams remaining after
2 such as 7.
1 21
2
approaches 0. each round is half the number of
1 teams that played in that round.
2 22
4
• If the tournament field was re-
3 23
8 3 2 1 O 1 2 3 x
7 duced to 32 teams, how many
basketball games would have
The domain is all real numbers, while the range is all positive numbers. to be played by the tourna-
ment’s winning team? 5 games
Lesson 10-1 Exponential Functions 523

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters Graphing Calculator and 5-Minute Check Transparency 10-1
• Study Guide and Intervention, pp. 573–574 Spreadsheet Masters, p. 45 Answer Key Transparencies
• Skills Practice, p. 575 Teaching Algebra With Manipulatives
• Practice, p. 576 Masters, pp. 276–277 Technology
• Reading to Learn Mathematics, p. 577 Interactive Chalkboard
• Enrichment, p. 578

Lesson x-x Lesson Title 523

 You can use a TI-83 Plus graphing calculator to look at the graph of two other

2 Teach exponential functions, y

3x and y
 .
1 x
3

EXPONENTIAL FUNCTIONS
Families of Exponential Functions
In-Class Example Power
Point® The calculator screen shows the graphs of
x
y
3x and y
 .
1
Teaching Tip Watch for stu- 3
dents who do not understand Think and Discuss 1–3,5. See margin. y ( 13 )
x
y  3x
why the graph in Example 1 1. How do the shapes of the graphs
cannot simply be modeled by a compare?
quadratic, cubic, or quartic 2. How do the asymptotes and
function. Point out that the y-intercepts of the graphs compare? [5, 5] scl: 1 by [2, 8] scl: 1

graphs of y
x2, y
x3, and 3. Describe the relationship between the
y
x4 all pass through the graphs.
point (0, 0) and not through the 4. See pp. 573A–573D. 4. Graph each group of functions on the same screen. Then compare the
point (0, 1). graphs, listing both similarities and differences in shape, asymptotes,
domain, range, and y-intercepts.
1 Sketch the graph of y
4x. a. y
2x, y
3x, and y
4x
Then state the function’s 1 x 1 x 1 x
b. y
 , y
 , and y

2 3 4
domain and range.
c. y
3(2)x and y
3(2)x ; y
1(2)x and y
2x.
y
5. Describe the relationship between the graphs of y
1(2)x and y
2x.

In general, an equation of the form y

abx, where a  0, b 0, and b  1, is called
Study Tip an exponential function with base b. Exponential functions have the following
Look Back characteristics.
To review continuous 1. The function is continuous and one-to-one.
O x functions, see page 63,
2. The domain is the set of all real numbers.
Exercises 60 and 61. To
review one-to-one 3. The x-axis is an asymptote of the graph.
functions, see Lesson 2-1.
The domain is all real numbers, 4. The range is the set of all positive numbers if a 0 and all negative
numbers if a  0.
while the range is all positive
5. The graph contains the point (0, a). That is, the y-intercept is a.
numbers.
1 x
6. The graphs of y
abx and y
a are reflections across the y-axis.
b
y
Answers Study Tip 4

Exponential There are two types of exponential functions: 3

1. The shapes of the graphs are the Growth and Decay exponential growth and exponential decay .
2
same. Notice that the graph of The base of an exponential growth function is
Exponential Exponential
an exponential growth a number greater than one. The base of an 1
2. The asymptote for each graph is function rises from left exponential decay function is a number
Decay Growth

the x-axis and the y-intercept for to right. The graph of between 0 and 1.
an exponential decay 2 1 O 1 2 x
each graph is 1. function falls from left
3. The graphs are reflections of each to right.

other over the y-axis.

5. The graphs are reflections of each Exponential Growth and Decay
other over the x-axis. • If a 0 and b 1, the function y
abx represents exponential growth.
• If a 0 and 0  b  1, the function y
abx represents exponential decay.

524 Chapter 10 Exponential and Logarithmic Relations

Families of Exponential Functions Have students begin by graphing the

two functions separately, so they recognize that the two curves shown in the
book are two distinct graphs. Students are used to seeing the U-shaped graphs
of polynomial functions and might have difficulty separating the graphs visually.
Also, a reminder about the meaning of the term asymptotes may be helpful for
many students.

524 Chapter 10 Exponential and Logarithmic Relations

 Example 2 Identify Exponential Growth and Decay In-Class Examples Power
Point®
Determine whether each function represents exponential growth or decay.
2 Determine whether each
Function Exponential Growth or Decay?
x
function represents
5
1
a. y
 The function represents exponential decay, since exponential growth or decay.
1
the base, , is between 0 and 1.
5
a. y
(0.7)x The function repre-
b. y
3(4)x The function represents exponential growth, since
sents exponential decay, since
the base, 4, is greater than 1.
the base, 0.7, is between 0 and 1.
c. y
7(1.2)x The function represents exponential growth, since
1
the base, 1.2, is greater than 1. b. y
(3)x The function repre-
2
sents exponential growth, since
the base, 3, is greater than 1.
x
Exponential functions are frequently used to model the growth or decay of a 3 4
c. y
10  The function repre-
population. You can use the y-intercept and one other point on the graph to write sents exponential growth, since
the equation of an exponential function. 4
the base, , is greater than 1.
3
Example 3 Write an Exponential Function 3 CELLULAR PHONES In
FARMING In 1983, there were 102,000 farms in Minnesota, but by 1998, this December of 1990, there were
number had dropped to 80,000. 5,283,000 cellular telephone
a. Write an exponential function of the form y  abx that could be used to subscribers in the United
model the farm population y of Minnesota. Write the function in terms of x, States. By December of 2000,
the number of years since 1983.
this number had risen to
For 1983, the time x equals 0, and the initial population y is 102,000. Thus, the 109,478, 000. Source: Cellular
y-intercept, and value of a, is 102,000. Telecommunications Industry Association

For 1998, the time x equals 1998  1983 or 15, and the population y is 80,000. a. Write an exponential function
Substitute these values and the value of a into an exponential function to of the form y
ab x that could
approximate the value of b. be used to model the number
Farming y
abx Exponential function of cellular telephone sub-
In 1999, 47% of the net 80,000
102,000b15 Replace x with 15, y with 80,000, and a with 102,000. scribers y in the U.S. Write
farm income in the United
0.78  b15 Divide each side by 102,000. the function in terms of x, the
States was from direct
government payments. The 
15
0.78  b Take the 15th root of each side. number of years since 1990.
USDA has set a goal of y  5,283,000(1.35)x
To find the 15th root of 0.78, use selection 5: 1 under the MATH menu on the
x
reducing this percent to
14% by 2005. TI-83 Plus. b. Suppose the number of
Source: USDA KEYSTROKES: 15 5 0.78 ENTER .9835723396 cellular telephone subscribers
continues to increase at the
An equation that models the farm population of Minnesota from 1983 to 1998 is same rate. Estimate the
TEACHING TIP y
102,000(0.98)x.
In Example 3, one of number of U.S. subscribers in
the given points is the b. Suppose the number of farms in Minnesota continues to decline at the same 2010. about 2,136,000,000
y-intercept. You may wish rate. Estimate the number of farms in 2010. subscribers
to give your students a
challenge problem in For 2010, the time x equals 2010  1983 or 27.
which any two points are
given and students use a y
102,000(0.98)x Modeling equation
system of equations to y
102,000(0.98)27 Replace x with 27. Interactive
find the equation of the y  59,115 Use a calculator.
exponential function. Chalkboard
The farm population in Minnesota will be about 59,115 in 2010. PowerPoint®
Presentations
www.algebra2.com/extra_examples Lesson 10-1 Exponential Functions 525
This CD-ROM is a customizable
Microsoft® PowerPoint®
presentation that includes:
Teacher to Teacher • Step-by-step, dynamic solutions of
David S. Daniels Longmeadow H.S., Longmeadow, MA each In-Class Example from the
Teacher Wraparound Edition
“As a lead-in activity for exponential functions, have students flip 50 pennies
and count the number of heads. Then have students remove those pennies • Additional, Your Turn exercises for
each example
that landed on heads and repeat the activity. Students should record their
results and make a plot of the trial number versus the number of heads • The 5-Minute Check Transparencies
1 x
counted in that trial. The graph will model that of y
 .” (2) • Hot links to Glencoe Online
Study Tools

Lesson 10-1 Exponential Functions 525

EXPONENTIAL EQUATIONS EXPONENTIAL EQUATIONS AND INEQUALITIES Since the domain of
AND INEQUALITIES Study Tip an exponential function includes irrational numbers such as 2
, all the properties
of rational exponents apply to irrational exponents.
Look Back
In-Class Examples Power
Point®
To review Properties of
Example 4 Simplify Expressions with Irrational Exponents
Power, see Lesson 5-1.

4 Simplify each expression. Simplify each expression.

a. 53  52 53  2 a. 25  23

b. (65 )6 630

 25  23
25  3 Product of Powers

5 Solve each equation. b. 723

2 723
72  3 Power of a Power
a. 49n  2
256 n  3
76 Product of Radicals
b. 35x
92x  1 x  2

The following property is useful for solving exponential equations. Exponential

equations are equations in which variables occur as exponents.

Property of Equality for Exponential Functions

• Symbols If b is a positive number other than 1, then bx
by
if and only if x
y.
• Example If 2x
28, then x
8.

Example 5 Solve Exponential Equations

Solve each equation.
a. 32n  1  81
32n  1
81 Original equation
32n  1
34 Rewrite 81 as 34 so each side has the same base.
2n  1
4 Property of Equality for Exponential Functions
2n
3 Subtract 1 from each side.
3
n
 Divide each side by 2.
2
3
The solution is .
2
CHECK 32n  1
81 Original equation

32   1  81
3 3
2 Substitute  for n.
2
34  81 Simplify.
81
81 ⻫ Simplify.

b. 42x  8x  1
42x
8x  1 Original equation
(22)2x
(23)x  1 Rewrite each side with a base of 2.
24x
23(x  1) Power of a Power
4x
3(x  1) Property of Equality for Exponential Functions
4x
3x  3 Distributive Property
x
3 Subtract 3x from each side.
The solution is 3.

526 Chapter 10 Exponential and Logarithmic Relations

526 Chapter 10 Exponential and Logarithmic Relations

 The following property is useful for solving inequalities involving exponential
functions or exponential inequalities . In-Class Example Power
Point®

Property of Inequality for Exponential Functions 6 Solve 53  2k 1. k  72

625
• Symbols If b  1, then bx  by if and only if x  y, and bx  by
if and only if x  y.
• Example If 5x  54, then x  4.

This property also holds for  and  . 3 Practice/Apply

Example 6 Solve Exponential Inequalities
1
Solve 43p  1  .
1
256
Study Notebook
43p  1  Original inequality
256 Have students—
43p  1 44 1 1
Rewrite  as 
256 4 or 4
4
4 so each side has the same base.
• add the definitions/examples of
3p  1 4 Property of Inequality for Exponential Functions the vocabulary terms to their
3p 3 Add 1 to each side. Vocabulary Builder worksheets for
p 1 Divide each side by 3. Chapter 10.
The solution set is p 1. • include examples of exponential
growth and decay graphs and
CHECK Test a value of p greater than 1; for example, p
0.
equations.
1
43p  1  Original inequality • include any other item(s) that they
256

43(0)  1
? 1
 Replace p with 0.
find helpful in mastering the skills
256
in this lesson.
? 1
41  Simplify.
256
1 1 1
  ⻫ a1

a
4 256

Answers
6. D  {x | x is all real numbers.},
R  {y | y  0}
Concept Check 1. OPEN ENDED Give an example of a value of b for which y
bx represents
2a. quadratic exponential decay. Sample answer: 0.8 y
2b. exponential 2. Identify each function as linear, quadratic, or exponential.
2c. linear a. y
3x2 b. y
4(3)x c. y
2x  4 d. y
4(0.2)x  1
2d. exponential Match each function with its graph.
1 x
3. y
5x c 4. y
2(5)x a 5. y
 b y  3(4)x
5
a. y b. y c. y

O x

7. D  {x | x is all real numbers.},

R  {y | y  0}
O x O x O x
y
Guided Practice Sketch the graph of each function. Then state the function’s domain and range.
1 x
6–7. See margin. 6. y
3(4)x 7. y
2
3
Lesson 10-1 Exponential Functions 527
( 13 )x
y2

O x
Differentiated Instruction
Auditory/Musical Going around the room, have students count by ones
beginning at 2, with each student calling out one number. Instruct them to
record the number they called as n. Then have students find n2 and 2n.
Now go around the room again and ask students to state their value of n2
(for a class of 30 students, the recited numbers are all the squares from 4
to 961). Now have students state their values of 2n (for a class of 30, the
recited numbers are all the powers of 2 from 4 to 231 or about 2 109).

Lesson 10-1 Exponential Functions 527

 GUIDED PRACTICE KEY Determine whether each function represents exponential growth or decay.
About the Exercises… Exercises Examples 8. y
2(7)x growth 9. y
(0.5)x decay 10. y
0.3(5)x growth
6, 7 1
Organization by Objective 8–10 2 Write an exponential function whose graph passes through the given points.
x
• Exponential Functions:
21–38, 57–61
11, 12,
19, 20
3  
11. (0, 3) and (1, 6) y  3 1
2
12. (0, 18) and (2, 2) y  18(3)x

13–15 4 Simplify each expression.

• Exponential Equations and 16–18 5, 6
Inequalities: 37–56, 62–66 13. 27  27 227 or 47 14. (a)4 a 4 15. 812  32
332 or 272
Odd/Even Assignments Solve each equation or inequality. Check your solution.
Exercises 21–56 are structured 1
16. 2n  4
 9 17. 52x  3  125 x
0 18. 92y  1
27y 2
so that students practice the 32
same concepts whether they Application ANIMAL CONTROL For Exercises 19 and 20, use the following information.
are assigned odd or even During the 19th century, rabbits were brought to Australia. Since the rabbits had no
problems. natural enemies on that continent, their population increased rapidly. Suppose there
were 65,000 rabbits in Australia in 1865 and 2,500,000 in 1867.
Alert! Exercise 61 involves
19. Write an exponential function that could be used to model the rabbit population
research on the Internet or y in Australia. Write the function in terms of x, the number of years since 1865.
other reference materials. y  65,000(6.20)x
20. Assume that the rabbit population continued to grow at that rate. Estimate the
Exercises 71–75 require the Australian rabbit population in 1872. 22,890,495,000
use of graphing calculators.
★ indicates increased difficulty
Assignment Guide Practice and Apply
Basic: 21, 23, 27–53 odd, 57–61, Homework Help Sketch the graph of each function. Then state the function’s domain
68–70, 76–89 For See and range. 21–26. See pp. 573A–573D.
Exercises Examples
Average: 21–55 odd, 59–64, 21–26 1
21. y
2(3)x 22. y
5(2)x 23. y
0.5(4)x
1 x
67–70, 76–89 (optional: 71–75) 27–32
33–38,
2
3  
24. y
4 
3
★ 25. y
 5 
1 x
★ 26. y
2.5(5)x
Advanced: 22–56 even, 61–86 57–66
39–44 4
(optional: 87–89) 45–56 5, 6
Determine whether each function represents exponential growth or decay.
1 x
27. y
10(3.5)x growth 28. y
2(4)x growth 29. y
0.4 decay
3
Extra Practice 5 x
See page 849. 30. y
3 growth 31. y
30x decay 32. y
0.2(5)x decay
2

Write an exponential function whose graph passes through the given points.
x
33. (0, 2) and (2, 32) y  2 1 
4
34. (0, 3) and (1, 15) y  3(5)x
1 x
35. (0, 7) and (2, 63) y  7(3) x 36. (0, 5) and (3, 135) y  5 
3  
37. (0, 0.2) and (4, 51.2) y  0.2(4)x 38. (0, 0.3) and (5, 9.6) y  0.3(2)x

Simplify each expression.

39. 528 54 or 625 
40. x53 x15 41. 72  732 742
42. y33  y3 y 23 43. n2  n n2  44. 64  2 25

Solve each equation or inequality. Check your solution. 54. p  2

2 1
45. 3n  2
27 5 46. 23x  5
128  47. 5n  3
 1
3 25
1 m 1 y3
49. 
81m  4  50. 
1 8
48. 22n   n
2
343 0
16 9 3 7
5
51. 16n  8n  1 n  3 52. 10x  1
1002x  3  53. 362p
216p  1 3
3
2
54. 325p  2  165p ★ 55. 35x  811  x
9x  3 10 56. 49x
7x  15 3, 5
528 Chapter 10 Exponential and Logarithmic Relations

Answers (p. 529)

60. 9.67 million; 17.62 million; 32.12 million; These answers are in close agreement with the
actual populations in those years.
61. 2144.97 million; 281.42 million; No, the growth rate has slowed considerably. The
population in 2000 was much smaller than the equation predicts it would be.

528 Chapter 10 Exponential and Logarithmic Relations

 BIOLOGY For Exercises 57 and 58, use the Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

10-1 Study Guide and

following information. p. 573 (shown)
Exponential Functionsand p. 574
The number of bacteria in a colony is
Log Exponential Functions An exponential function has the form y
abx,
The magnitude of an growing exponentially. where a  0, b 0, and b  1.
umber of
earthquake can be Time N ria 1. The function is continuous and one-to-one.

Bacte
2. The domain is the set of all real numbers.

represented by an
57. Write an exponential function to model Properties of an
Exponential Function
3.
4.
The
The
x-axis is the asymptote of the graph.
range is the set of all positive numbers if a 0 and all negative numbers if a  0.

the population y of bacteria x hours 100 5. The graph contains the point (0, a).

exponential equation. 2 P.M. If a 0 and b 1, the function y

abx represents exponential growth.
4 000
Exponential Growth

after 2 P.M. y  100(6.32)x

Lesson 10-1
and Decay If a 0 and 0  b  1, the function y
abx represents exponential decay.
4 P. M .
Visit www.algebra2. Example 1 Sketch the graph of y  0.1(4)x. Then state the y
com/webquest to 58. How many bacteria were there at 7 P.M. function’s domain and range.
Make a table of values. Connect the points to form a smooth curve.
continue work on your that day? about 1,008,290 x 1 0 1 2 3

WebQuest project. y 0.025 0.1 0.4 1.6 6.4

O x
The domain of the function is all real numbers, while the range is
the set of all positive real numbers.

POPULATION For Exercises 59–61, use the following information. Example 2 Determine whether each function represents exponential
growth or decay.
Every ten years, the Bureau of the Census counts the number of people living a. y
0.5(2) x b. y
2.8(2) x c. y
1.1(0.5) x

in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, exponential growth,
since the base, 2, is
neither, since 2.8,
the value of a is less
exponential decay, since
the base, 0.5, is between

this number had grown to 5.31 million. greater than 1 than 0. 0 and 1

Exercises
Sketch the graph of each function. Then state the function’s domain and range.
59. Write an exponential function that could be used to model the U.S. population y 1. y
3(2) x 2. y
2   14 
x
3. y
0.25(5) x

in millions for 1790 to 1800. Write the equation in terms of x, the number of y y y

decades x since 1790. y  3.93(1.35)x O x

62. Exponential; the 60. Assume that the U.S. population continued to grow at that rate. Estimate the
r O x O x

base, 1  , is population for the years 1820, 1840, and 1860. Then compare your estimates Domain: all real Domain: all real Domain: all real
n numbers; Range: all numbers; Range: all numbers; Range: all
fixed, but the expo- with the actual population for those years, which were 9.64, 17.06, and positive real numbers negative real numbers positive real numbers

nent, nt, is variable 31.44 million, respectively. See margin. Determine whether each function represents exponential growth or decay.
x
4. y
0.3(1.2) x growth 5. y
5   45  neither 6. y
3(10)x decay

since the time t can 61. RESEARCH Estimate the population of the U.S. in 2000. Then use the Internet Gl NAME
/M G ______________________________________________
Hill 573 DATE ____________
GlPERIOD
Al _____
b 2

vary. Skills
10-1 Practice,
Practice p. 575 and
or other reference to find the actual population of the U.S. in 2000. Has the (Average)
Practice, p.Functions
Exponential 576 (shown)
population of the U.S. continued to grow at the same rate at which it was Sketch the graph of each function. Then state the function’s domain and range.
growing in the early 1800s? Explain. See margin. 1. y
1.5(2)x 2. y
4(3)x 3. y
3(0.5)x
y y y

MONEY For Exercises 62–64, use the following information.

Suppose you deposit a principal amount of P dollars in a bank account that pays O x O x O x

compound interest. If the annual interest rate is r (expressed as a decimal) and the domain: all real domain: all real domain: all real
numbers; range: all numbers; range: all numbers; range: all
bank makes interest payments n times every year, the amount of money A you positive numbers positive numbers positive numbers

r nt
would have after t years is given by A(t)
P1   .
Determine whether each function represents exponential growth or decay.
4. y
5(0.6) x decay 5. y
0.1(2) x growth 6. y
5  4x decay
n Write an exponential function whose graph passes through the given points.
9. (0, 3) and (1, 1.5)
62. If the principal, interest rate, and number of interest payments are known, 7. (0, 1) and (1, 4) 8. (0, 2) and (1, 10)

 41 
x
y  y  2(5)x y  3(0.5)x
r nt
 n 
what type of function is A(t)
P 1   ? Explain your reasoning. 10. (0, 0.8) and (1, 1.6)
y  0.8(2)x
11. (0, 0.4) and (2, 10)
y  0.4(5)x
12. (0, ) and (3, 8)
y  (2)x

Computers 63. Write an equation giving the amount of money you would have after Simplify each expression.
13. (22)8 16 14. (n3)
75 n15 15. y6  y56 y 66

Since computers were

t years if you deposit $1000 into an account paying 4% annual interest 24 133
16. 136  13 6 17. n3  n n 3  18. 125 11 52
11  5 11

invented, computational compounded quarterly (four times per year). A(t)  1000(1.01)4t Solve each equation or inequality. Check your solution.
1
19. 33x  5 81 x  3 20. 76x
72x  20 5 21. 36n  5  94n  3 n  
speed has multiplied by a 2
64. Find the account balance after 20 years. $2216.72 22. 92x  1
27x  4 14 23. 23n  1    18 
n
n
1
24. 164n  1
1282n  1 
11
factor of 4 about every 6 2
BIOLOGY For Exercises 25 and 26, use the following information.
three years. The initial number of bacteria in a culture is 12,000. The number after 3 days is 96,000.

Source: www.wired.com COMPUTERS For Exercises 65 and 66, use the information at the left. 25. Write an exponential function to model the population y of bacteria after x days.
y  12,000(2)x
26. How many bacteria are there after 6 days? 768,000
65. If a typical computer operates with a computational speed s today, write an 27. EDUCATION A college with a graduating class of 4000 students in the year 2002
expression for the speed at which you can expect an equivalent computer to predicts that it will have a graduating class of 4862 in 4 years. Write an exponential
function to model the number of students y in the graduating class t years after 2002.
operate after x three-year periods. s  4x y  4000(1.05)t

Gl NAME
/M G ______________________________________________
Hill 576 DATE ____________
Gl PERIOD
Al _____
b 2

★ 66. Suppose your computer operates with a processor speed of 600 megahertz and Reading
10-1 Readingto
to Learn
Learn Mathematics
Mathematics, p. 577 ELL
you want a computer that can operate at 4800 megahertz. If a computer with Exponential Functions
Pre-Activity How does an exponential function describe tournament play?
that speed is currently unavailable for home use, how long can you expect to Read the introduction to Lesson 10-1 at the top of page 523 in your textbook.

wait until you can buy such a computer? 1.5 three-year periods or 4.5 yr How many rounds of play would be needed for a tournament with 100
players? 7

Reading the Lesson

67. Sometimes; true ★ 67. CRITICAL THINKING Decide whether the following statement is sometimes,

Lesson 10-1
1. Indicate whether each of the following statements about the exponential function

when b  1, but false always, or never true. Explain your reasoning. y

10x is true or false.

a. The domain is the set of all positive real numbers. false

when b  1. For a positive base b other than 1, bx by if and only if x y. b. The y-intercept is 1. true
c. The function is one-to-one. true

www.algebra2.com/self_check_quiz Lesson 10-1 Exponential Functions 529 d. The y-axis is an asymptote of the graph. false

e. The range is the set of all real numbers. false

2. Determine whether each function represents exponential growth or decay.

 25 
x
a. y
0.2(3) x. growth b. y
3  . decay c. y
0.4(1.01) x. growth
NAME ______________________________________________ DATE ____________ PERIOD _____
3. Supply the reason for each step in the following solution of an exponential equation.
92x  1
27x
Enrichment,
10-1 Enrichment p. 578 (32)2x  1
(33) x
Original equation
Rewrite each side with a base of 3.
32(2x  1)
33x Power of a Power
Finding Solutions of xy  yx 2(2x  1)
3x Property of Equality for Exponential Functions
4x  2
3x Distributive Property
Perhaps you have noticed that if x and y are interchanged in equations such
as x
y and xy
1, the resulting equation is equivalent to the original x2
0 Subtract 3x from each side.
equation. The same is true of the equation x y
y x. However, finding x
2 Add 2 to each side.
solutions of x y
y x and drawing its graph is not a simple process.

Solve each problem. Assume that x and y are positive real numbers. Helping You Remember
1. If a 0, will (a, a) be a solution of x y
y x? Justify your answer. 4. One way to remember that polynomial functions and exponential functions are different
is to contrast the polynomial function y
x2 and the exponential function y
2x. Tell at
Yes, since aa  aa must be true (Reflexive Prop. of Equality). least three ways they are different.
Sample answer: In y  x 2, the variable x is a base, but in y  2x, the
2. If c 0, d 0, and (c, d) is a solution of xy
yx, will (d, c) also
variable x is an exponent. The graph of y  x 2 is symmetric with respect
be a solution? Justify your answer.
to the y-axis, but the graph of y  2x is not. The graph of y  x 2 touches
the x-axis at (0, 0), but the graph of y  2x has the x-axis as an asymptote.
Yes; replacing x with d, y with c gives d c  c d; but if (c, d) is a solution, You can compute the value of y  x 2 mentally for x  100, but you cannot
c d  d c. So, by the Symmetric Property of Equality, d c  c d is true. compute the value of y  2x mentally for x  100.

3. Use 2 as a value for y in x y

y x. The equation becomes x 2
2 x.

Lesson 10-1 Exponential Functions 529

 68. WRITING IN MATH Answer the question that was posed at the beginning of
4 Assess the lesson. See pp. 573A–573D.
How does an exponential function describe tournament play?
Open-Ended Assessment Include the following in your answer:
• an explanation of how you could use the equation y
2x to determine the
Modeling Give students a sheet number of rounds of tournament play for 128 teams, and
of grid paper and a length of
• an example of an inappropriate number of teams for tournament play with
string. Have students model the an explanation as to why this number would be inappropriate.
 21 
x
graph of the equation y
 .
Have them check their model by Standardized 69. If 4x  2
48, then 4x
A
graphing the equation on a Test Practice A 3.0. B 6.4. C 6.9. D 12.0. E 24.0.
graphing calculator.
70. GRID IN Suppose you deposit $500 in an account paying 4.5% interest
compounded semiannually. Find the dollar value of the account rounded to
Getting Ready for the nearest penny after 10 years. 780.25
Lesson 10-2
PREREQUISITE SKILL In Lesson
Graphing FAMILIES OF GRAPHS Graph each pair of functions on the same screen. Then
10-2, students will evaluate compare the graphs, listing both similarities and differences in shape,
Calculator
logarithmic expressions. Because asymptotes, domain, range, and y-intercepts. 7174. See pp. 573A–573D.
logarithmic and exponential 71. y
2x and y
2x  3 72. y
3x and y
3x  1
functions are inverses of each x x2 1 x 1 x
73. y
 and y
 74. y
 and y
  1
1 1
other, their composites are the 5 5 4 4
identity function. Students must 75. Describe the effect of changing the values of h and k in the equation
be familiar with compositions of y
2x  h  k. See margin.
functions in order to evaluate
these inverse functions. Use
Exercises 87–89 to determine Maintain Your Skills
your students’ familiarity with
composition of functions. Mixed Review Solve each equation or inequality. Check your solutions. (Lesson 9-6)
15 s3 6
76.   p
16 1, 15 77. 

2  1, 6
p s4 s  16

Answers 2a  5 a 6 13 x2 x4

78.   
  , 3 79.    0  x  3 or x  6
a9 a9 a2  81 3 x x6
75. For h  0, the graph of y  2x is
translated | h | units to the right.
Identify each equation as a type of function. Then graph the equation. (Lesson 9-5)

For h  0, the graph of y  2x is 80–82. See margin 80. y


x2 81. y
2x 82. y
8
for graphs. square root greatest integer constant
translated | h | units to the left. For Find the inverse of each matrix, if it exists. (Lesson 4-7)
k  0, the graph of y  2x is 1 0 5 6 1 3 –6
0 1
0 1  
1 0 2 4
translated | k | units up. For k  0, 83. 84. 5 10

does not exist 85.

11 3
51 11

–5
the graph of y  2x is translated
| k | units down. 86. ENERGY A circular cell must deliver 18 watts of energy. If each square
centimeter of the cell that is in sunlight produces 0.01 watt of energy, how long
80. y must the radius of the cell be? (Lesson 5-8) about 23.94 cm
y  x  2

Getting Ready for PREREQUISITE SKILL Find g[h(x)] and h[g(x)].

the Next Lesson (To review composition of functions, see Lesson 7-7.) 87–89. See margin.
O x 87. h(x)
2x  1 88. h(x)
x  3 89. h(x)
2x  5
g(x)
x  5 g(x)
x2 g(x)
x  3
530 Chapter 10 Exponential and Logarithmic Relations

81. y
82. y 87. g[h(x)]  2x  6; h[g(x)]  2x  11
y  2x 
88. g[h(x)]  x 2  6x  9; h[g(x)]  x 2  3
y8 89. g[h(x)]  2x  2; h[g(x)]  2x  11
O x

O x

530 Chapter 10 Exponential and Logarithmic Relations

 Logarithms and Lesson
Logarithmic Functions Notes

• Evaluate logarithmic expressions.

• Solve logarithmic equations and inequalities.
1 Focus
Vocabulary is a logarithmic scale used to measure sound?
• logarithm Many scientific measurements have such an enormous range of possible values
5-Minute Check
• logarithmic function that it makes sense to write them as powers of 10 and simply keep track of their Transparency 10-2 Use as
• logarithmic equation exponents. For example, the loudness of sound is measured in units called a quiz or review of Lesson 10-1.
• logarithmic inequality decibels. The graph shows the relative intensities and decibel measures of
common sounds. Mathematical Background notes
are available for this lesson on
Sound p. 520C.

Relative 0 2 4 6 8 10 12
is a logarithmic scale
10 10 10 10 10 10 10
Intensity used to measure sound?
Decibels 0 20 40 60 80 100 120 Ask students:
pin whisper normal noisy
drop (4 feet) conversation kitchen jet engine • On the number line shown, the
The decibel measure of the loudness of a sound is the exponent or logarithm of scale along the bottom is 10
its relative intensity multiplied by 10. decibels per tick mark. What do
you notice about the scale along
the top for relative intensity?
LOGARITHMIC FUNCTIONS AND EXPRESSIONS To better understand The scale is not uniform; the
what is meant by a logarithm, let’s look at the graph of y
2x and its inverse. Since relative intensity at the first tick
Study Tip exponential functions are one-to-one, the inverse of y
2x exists and is also a
function. Recall that you can graph the inverse of a function by interchanging the mark is 10, at the second it is 100,
Look Back
To review inverse x and y values in the ordered pairs of the function. at the third it is 1000, and so on.
functions, see Lesson 7-8.
• If you draw a number line with
y  2x x  2y y a uniform scale whose tick
x y x y y  2x marks are labeled from 0 to 1012,
3
1

1
 3 what number is at the midpoint
8 8
(2, 4) yx between 0 to 1012? 5  1011
1 1
2   2
4 4 • Where does the point 1 million
1
1

1
 1 (0, 1) (4, 2) x  2y
appear on your number line?
2 2
O x
very close to the point for 0
0 1 1 0 (1, 0)

1 2 2 1
• Where does the point 100
As the value of y
2 4 4 2 decreases, the value appear on your number line?
3 8 8 3
of x approaches 0. very, very close to the point for 0
• What problem arises with trying
The inverse of y
2x can be defined as x
2y. Notice that the graphs of
to represent the relative intensi-
these two functions are reflections of each other over the line y
x. ties on a standard number line?
In general, the inverse of y
bx is x
by. In x
by, y is called the Sample answer: The lesser intensi-
logarithm of x. It is usually written as y
logb x and is read y equals ties are so close together near 0 on
log base b of x. the number line that they are
Lesson 10-2 Logarithms and Logarithmic Functions 531 difficult to represent accurately.

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters School-to-Career Masters, p. 19 5-Minute Check Transparency 10-2
• Study Guide and Intervention, pp. 579–580 Answer Key Transparencies
• Skills Practice, p. 581
• Practice, p. 582 Technology
• Reading to Learn Mathematics, p. 583 Interactive Chalkboard
• Enrichment, p. 584
• Assessment, p. 623

Lesson x-x Lesson Title 531

 Logarithm with Base b
2 Teach • Words Let b and x be positive numbers, b  1. The logarithm of x with base b
is denoted logb x and is defined as the exponent y that makes the
equation by
x true.
LOGARITHMIC FUNCTIONS • Symbols Suppose b 0 and b  1. For x 0, there is a number y such that
AND EXPRESSIONS logb x
y if and only if by
x.

Teaching Tip After discussing the

definition of logarithm at the bottom Example 1 Logarithmic to Exponential Form
of p. 531, write the equation y
2x Write each equation in exponential form.
on the chalkboard and ask students 1
to rewrite the equation with x in a. log8 1  0 b. log2   4
16

 1
2 
terms of y. x
y Repeat this for log8 1
0 → 1
80
1
16
1
log2 
4 → 
24
16
the equation y
x2. (x
y )
Now write the equation y
2x on
the chalkboard and ask students to Example 2 Exponential to Logarithmic Form
rewrite this equation with x in terms Write each equation in logarithmic form.
of y. This will likely have students 1

stymied. Explain that the rewritten a. 103  1000 b. 9 2  3
equation is x
log2 y. 1
 1
103
1000 → log10 1000
3 9 2
3 → log9 3

2

In-Class Examples Power

Point®
You can use the definition of logarithm to find the value of a logarithmic
expression.
1 Write each equation in
exponential form. Example 3 Evaluate Logarithmic Expressions
a. log3 9
2 32 9 Evaluate log2 64.
1 1 log2 64
y
b. log10 
2 102   
100
Let the logarithm equal y.
100 64
2y Definition of logarithm
2 Write each equation in 26
2y 64
26
logarithmic form. 6
y Property of Equality for Exponential Functions
a. 53
125 log5 125  3 So, log2 64
6.
1
 1
b. 27 3
3 log27 3  3
The function y
logb x, where b 0 and b  1, is called a logarithmic function..
3 Evaluate log3 243. 5 As shown in the graph on the previous page, this function is the inverse of the
exponential function y
bx and has the following characteristics.
1. The function is continuous and one-to-one.
2. The domain is the set of all positive real numbers.
3. The y-axis is an asymptote of the graph.
4. The range is the set of all real numbers.
Study Tip 5. The graph contains the point (1, 0). That is, the x-intercept is 1.
Look Back Since the exponential function f(x)
bx and the logarithmic function g(x)
logb x
To review composition of are inverses of each other, their composites are the identity function. That is,
functions, see Lesson 7-7. f[g(x)]
x and g[f(x)]
x.
f[g(x)]
x g[ f(x)]
x
f(logb x)
x g(bx)
x
blogb x
x logb bx
x
532 Chapter 10 Exponential and Logarithmic Relations

532 Chapter 10 Exponential and Logarithmic Relations

 Thus, if their bases are the same, exponential and logarithmic functions “undo”
each other. You can use this inverse property of exponents and logarithms to In-Class Example Power
Point®
simplify expressions.
4 Evaluate each expression.
Example 4 Inverse Property of Exponents and Logarithms
a. log9 92 2
Evaluate each expression.
b. 7log 7 (x
2  1) x2  1
a. log6 68 b. 3log3 (4x  1)

log6 68
8 logb bx
x 3log3 (4x – 1)
4x  1 blogb x
x

SOLVE LOGARITHMIC
SOLVE LOGARITHMIC EQUATIONS AND INEQUALITIES A EQUATIONS AND
logarithmic equation is an equation that contains one or more logarithms. You can INEQUALITIES
use the definition of a logarithm to help you solve logarithmic equations.
In-Class Examples Power
Point®
Example 5 Solve a Logarithmic Equation
5 5 Solve log8 n
43. 16
Solve log4 n  .
2
5
log4 n
 Original equation 6 Solve log6 x 3. Check your
solution. {x | x  216}
2
5

n
42 Definition of logarithm
5

n
(22) 2 4 = 22

n
25 Power of a Power

n
32 Simplify.

A logarithmic inequality is an inequality that involves logarithms. In the case of

inequalities, the following property is helpful.

Logarithmic to Exponential Inequality

• Symbols If b 1, x 0, and logb x y, then x by.
If b 1, x 0, and logb x  y, then 0  x  by.
• Examples log2 x 3 log3 x  5
x 23 0  x  35

Example 6 Solve a Logarithmic Inequality

Solve log5 x  2. Check your solution.
log5 x  2 Original inequality
Study Tip 0x 52 Logarithmic to exponential inequality
Special Values 0  x  25 Simplify.
If b > 0 and b ≠ 1, then
the following statements The solution set is {x0  x  25}.
are true.
• logb b
1 because
b1
b.
CHECK Try 5 to see if it satisfies the inequality.
• logb 1
0 because log5 x  2 Original inequality
b0
1. ?
log5 5  2 Substitute 5 for x.

12⻫ log55
1 because 51
5.

www.algebra2.com/extra_examples Lesson 10-2 Logarithms and Logarithmic Functions 533

Lesson 10-2 Logarithms and Logarithmic Functions 533

 Use the following property to solve logarithmic equations that have logarithms
In-Class Examples Power
Point® with the same base on each side.

7 Solve log4 x2
log4 (4x  3). Property of Equality for Logarithmic Functions
Check your solution. 1, 3 • Symbols If b is a positive number other than 1, then
logb x
logb y if and only if x
y.
8 Solve • Example If log7 x
log7 3, then x
3.
log7 (2x  8) log7 (x  5).
Check your solution. x  3
Example 7 Solve Equations with Logarithms on Each Side
Solve log5 (p2  2)  log5 p. Check your solution.
log5 (p2  2)
log5 p Original equation
Intervention
p2  2
p Property of Equality for Logarithmic Functions
New Students have
p2 p2
0 Subtract p from each side.
not covered
(p  2)(p  1)
0 Factor.
logarithmic
p2
0 or p  1
0 Zero Product Property
functions before
and are likely to find them p
2 p
1 Solve each equation.
confusing. Expect students to Study Tip CHECK Substitute each value into the original equation.
need extra time to absorb the log5 (22  2)  log5 2 Substitute 2 for p.
Extraneous
material in this lesson before Solutions log5 2
log5 2 ⻫ Simplify.
continuing with the rest of the The domain of a
logarithmic function does log5 [(1)2  2]  log5 (1) Substitute 1 for p.
chapter. not include negative Since log5 (1) is undefined, 1 is an extraneous solution and must be
values. For this reason,
eliminated. Thus, the solution is 2.
be sure to check for
extraneous solutions of
logarithmic equations.
Use the following property to solve logarithmic inequalities that have the same
base on each side. Exclude values from your solution set that would result in taking
the logarithm of a number less than or equal to zero in the original inequality.

Property of Inequality for Logarithmic Functions

• Symbols If b 1, then logb x logb y if and only if x y, and
logb x  logb y if and only if x  y.
• Example If log2 x log2 9, then x 9.

This property also holds for  and .

Example 8 Solve Inequalities with Logarithms on Each Side

Solve log10 (3x  4)  log10 (x  6). Check your solution.
log10 (3x  4)  log10 (x  6) Original inequality
3x  4  x  6 Property of Inequality for Logarithmic Functions

2x  10 Addition and Subtraction Properties of Inequalities

x5 Divide each side by 2.

Study Tip
We must exclude from this solution all values of x such that 3x  4  0 or x  6  0.
Look back 4
To review compound Thus, the solution set is x  and x 6 and x  5. This compound inequality
3
4
inequalities, see simplifies to   x  5.
Lesson 1-6. 3

534 Chapter 10 Exponential and Logarithmic Relations

Differentiated Instruction
Visual/Spatial Have students create colorful posters showing several
equivalent exponential and logarithmic equations, such as 23
8 and
3
log2 8. Suggest that students use a different color for each of the
digits 2, 3, and 8 to help them visualize the relative locations of the
digits in the pairs of equations.

534 Chapter 10 Exponential and Logarithmic Relations

 Concept Check 1. OPEN ENDED Give an example of an exponential equation and its related
3 Practice/Apply
logarithmic equation. Sample answer: x  5y and y  log5 x
2. Describe the relationship between y
3x and y
log3 x. They are inverses.
3. FIND THE ERROR Paul and Scott are solving log3 x
9. Study Notebook
Paul Scot t Have students—
• add the definitions/examples of
log3 x = 9 log 3 x = 9
the vocabulary terms to their
x
3 =9 x = 39 Vocabulary Builder worksheets for
x
3 =3
2
x = 19,683
Chapter 10.
• include examples of how to write
x=2 logarithms in exponential form.
Who is correct? Explain your reasoning. Scott; see margin for explanation.
• include any other item(s) that they
find helpful in mastering the skills
Guided Practice Write each equation in logarithmic form. in this lesson.
1 1
GUIDED PRACTICE KEY 4. 54
625 log5 625  4 5. 72
 log7   2
49 49
Exercises Examples
Write each equation in exponential form. 1
4, 5 1 1 
6, 7 2 6. log3 81
4 34  81 7. log36 6
 36 2  6
2
8–11 3
12–17 4–7 Evaluate each expression. FIND THE ERROR
18–20 4 1 Review convert-
8. log4 256 4 9. log2  3 10. 3log3 21 21 11. log5 51 1
8 ing logarithms to
Solve each equation or inequality. Check your solutions. exponential form. Also note that,
3
12. log9 x
 27 13. log1 x
3 1000
according to Paul, log3 x
3x,
2 10 which cannot be true.
1 1
14. log3 (2x  1)  2   x
5 15. log5 (3x  1)
log5 2x2 , 1
2 2
16. log2 (3x  5) log2 (x  7) x  6 17. logb 9
2 3
Answer
Application SOUND For Exercises 18–20, use 3. The value of a logarithmic
the following information. USA TODAY Snapshots® equation, 9, is the exponent of the
An equation for loudness L, in equivalent exponential equation,
decibels, is L
10 log10 R, where R July 4th can be loud. Be careful.
is the relative intensity of the sound. Any sound above 85 decibels has the potential and the base of the logarithmic
to damage hearing. The noisiest Fourth of July
activities, in decibels: expression, 3, is the base of the
18. Solve 130
10 log10 R to
find the relative intensity exponential equation. Thus x  39
of a fireworks display with Fireworks 130-190
or 19,683.
a loudness of 130 decibels. 1013
Car racing 100-130
19. Solve 75
10 log10 R to find Parades 80-120
the relative intensity of a Yard work 95-115
concert with a loudness of
Movies 90-110
75 decibels. 107.5
Concerts 75-110
20. How many times more intense
is the fireworks display than
the concert? In other words, find Note: Sounds listed by range of peak levels.

the ratio of their intensities. 105.5 Source: National Campaign for Hearing Health

or about 316,228 times By Hilary Wasson and Sam Ward, USA TODAY

Lesson 10-2 Logarithms and Logarithmic Functions 535

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interactive features connected to each day’s newspaper.
Experience TODAY, USA TODAY’s daily lesson plan, is
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This plan provides instruction for integrating USA TODAY
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classroom. Log on to www.education.usatoday.com.

Lesson 10-2 Logarithms and Logarithmic Functions 535

 ★ indicates increased difficulty
About the Exercises… Practice and Apply
Organization by Objective 1
Homework Help Write each equation in logarithmic form. 23. log5   3
125
• Logarithmic Functions and For See
21. 83
512 log8 512  3 22. 33
27 log3 27  3
1
23. 53

Exercises Examples 125
Expressions: 21–46, 66–71 1 1
21–26 1 1 2  
• Solve Logarithmic Equations 27–32 2 3 
24. 
9 log1 9  2 25. 100
10 1
3
2

log100 10  
26. 2401
7 4

log2401 7  
1
33–46 3
and Inequalities: 47–65 2 4
47–62 4–7 1
63–65 4 Write each equation in exponential form. 29. 41  
Odd/Even Assignments 4 1
68–70 5 27. log5 125
3 5  125 28. log13 169
2 13  169 29. log4 
1
3 2
Exercises 21–62 are structured 2

4
Extra Practice 1 1 2
so that students practice the 30. log100 

10 2
31. log8 4

3
83 4 32. log1 25
2
See page 849. 5
same concepts whether they 1 1
100 2  10  
2
1  25
are assigned odd or even Evaluate each expression. 5
1
problems. 33. log2 16 4 34. log12 144 2 35. log16 4 
2
5 1 1
36. log9 243  37. log2  5 38. log3  4
Assignment Guide 2 32 81
39. log5 57 7 40. 2log2 45 45 41. log11 11(n  5) n  5
Basic: 21–41 odd, 45–59 odd,
71–90 42. 6log6 (3x  2) 3x  2 ★ 43. log10 0.001 3 ★ 44. log4 16x 2x
Average: 21–67 odd, 68, 69,
71–90 WORLD RECORDS For Exercises 45 and 46, use the information given for
Exercises 18–20 to find the relative intensity of each sound. Source: The Guinness Book of Records
Advanced: 22–66 even, 68–84
(optional: 85–90) 45. The loudest animal sounds are the 46. The loudest insect is the African
low-frequency pulses made by blue cicada. It produces a calling song
All: Practice Quiz 1 (1–10) whales when they communicate. that measures 106.7 decibels at a
These pulses have been measured distance of 50 centimeters. 1010.67
up to 188 decibels. 1018.8

Solve each equation or inequality. Check your solutions.

47. log9 x
2 81 48. log2 c 8 c  256
1 3
49. log64 y  
2
0y
8 50. log25 n
 125
2
51. log1 x
1 7 52. log1 p  0 0  p  1
7 3
53. log2 (3x  8)  6 x  24 54. log10 (x2  1)
1 3

55. logb 64
3 4 56. logb 121
2 11

57. log5 56n  1

13 2
1
58. log5 x

2
5
59. log6 (2x  3)
log6 (x  2) 5 60. log2 (4y  10)  log2 (y  1) y  3
Answers ★ 61. log10 (a2  6) log10 a a  3 ★ 62. log7 (x2  36)
log7 100 8
63. log5 25  2 log5 5 Original
equation Show that each statement is true. 63–65. See margin.
log5 52  2 log5 51 25  52 and ★ 63. log5 25
2 log5 5 ★ 64. log16 2  log2 16
1 ★ 65. log7 [log3 (log2 8)]
0
5  51
536 Chapter 10 Exponential and Logarithmic Relations
2  2(1) Inverse
Property of
Exponents and
Logarithms 65. log7 [log3 (log2 8)] 0 Original equation
22✓ Simplify.
log7 [log3 (log2 23)] 0 8  23
64. log16 2  log2 16  1 Original
1 equation log7 (log3 3) 0 Inverse Property of Exponents and Logarithms
 1
log16 16  log2 24  log7 (log3 31) 0

4
1 2  16 4 and 3  31
1 16  2 4
(4)  1 Inverse log7 1 0 Inverse Property of Exponents and Logarithms
4 Property of log7 70 0 1  70
Exponents and
Logarithms 0 0✓ Inverse Property of Exponents and Logarithms
1 1 ✓

536 Chapter 10 Exponential and Logarithmic Relations

 1 x
66. a. Sketch the graphs of y
log1 x and y
 on the same axes.
NAME ______________________________________________ DATE ____________ PERIOD _____

66–67. Study
10-2 Guide
Study andIntervention
Guide and Intervention,
2 2 p. 579 (shown) and Functions
p. 580
See pp. 573A–573D. Logarithms and Logarithmic
b. Describe the relationship between the graphs. Logarithmic Functions and Expressions
Definition of Logarithm Let b and x be positive numbers, b  1. The logarithm of x with base b is denoted
with Base b logb x and is defined as the exponent y that makes the equation b y
x true.

★ 67. a. Sketch the graphs of y

log2 x  3, y
log2 x  4, y
log2 (x  1), and The inverse of the exponential function y
bx is the logarithmic function x
by.
This function is usually written as y
logb x.

y
log2 (x  2). 1.
2.
The
The
function is continuous and one-to-one.
domain is the set of all positive real numbers.
Properties of
3. The y-axis is an asymptote of the graph.
Logarithmic Functions

b. Describe this family of graphs in terms of its parent graph y

log2 x. 4.
5.
The
The
range is the set of all real numbers.
graph contains the point (0, 1).

Example 1 Write an exponential equation equivalent to log3 243  5.

35
243

EARTHQUAKE For Exercises 68 and 69, use the following information.

Lesson 10-2
Example 2 Write a logarithmic equation equivalent to 63   .
1
216
1
The magnitude of an earthquake is measured on a logarithmic scale called the log6 
3
216

Richter scale. The magnitude M is given by M

log10 x, where x represents the Example 3
4
Evaluate log8 16.
4
8 3
16, so log8 16
 .
amplitude of the seismic wave causing ground motion. 3

Exercises
68. How many times as great is the amplitude caused by an earthquake with a Write each equation in logarithmic form.

Richter scale rating of 7 as an aftershock with a Richter scale rating of 4? 1. 27

128 2. 34

1
81
3.   17 
3


1
343

103 or 1000 as times great log2 128  7 log3   4

1
81
log17   3
1
343
69. How many times as great was the motion caused by the 1906 San Francisco Write each equation in exponential form.
earthquake that measured 8.3 on the Richter scale as that caused by the 2001 4. log15 225
2 5. log3 
3
1
27
6. log4 32

5
2
Bhuj, India, earthquake that measured 6.9? 101.4 or about 25 times as great 152  225 33  
1
27
5

4 2  32

Evaluate each expression.

7. log4 64 3 8. log2 64 6 9. log100 100,000 2.5

70. NOISE ORDINANCE A proposed city ordinance will make it illegal to 10. log5 625 4 11. log27 81 
4
12. log25 5 
1
3 2
create sound in a residential area that exceeds 72 decibels during the day 13. log2  7
1
14. log10 0.00001 5 15. log4  2.5
1
128 32
and 55 decibels during the night. How many times more intense is the
noise level allowed during the day than at night? 101.7 or about 50 times Gl NAME
/M G ______________________________________________
Hill 579 DATE ____________
GlPERIOD
Al _____
b 2

Skills
10-2 Practice,
Practice (Average)
p. 581 and
Earthquake Practice,
Logarithmsp.
and582 (shown)
Logarithmic Functions
Write each equation in logarithmic form.
The Loma Prieta 71. CRITICAL THINKING The value of log2 5 is between two consecutive 1. 53
125 log5 125  3 2. 70
1 log7 1  0 3. 34
81 log3 81  4

earthquake measured 7.1 integers. Name these integers and explain how you determined them. 1
 14 
1 3 1
4. 34
 5. 
 6. 7776 5
6
81 64
on the Richter scale and 2 and 3; Sample answer: 5 is between 22 and 23. 1
log3   4
81
1
log   3
64
1

4
log7776 6  
1
5
interrupted the 1989 World
Series in San Francisco. 72. CRITICAL THINKING Using the definition of a logarithmic function where Write each equation in exponential form.

Source: U.S. Geological Survey

y
logb x, explain why the base b cannot equal 1. All powers of 1 are 1, so the 7. log6 216
3 63  216 8. log2 64
6 26  64 9. log3 
4 34  
1
81
1
81

inverse of y  1x is not a function. 10. log10 0.00001

5 11. log25 5

1

1
2
12. log32 8

3

3
5

105  0.00001 25 2  5 32 5  8
73. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See pp. 573A–573D. Evaluate each expression.
1
13. log3 81 4 14. log10 0.0001 4 15. log2  4 16. log13 27 3
16
Why is a logarithmic scale used to measure sound? 17. log9 1 0 18. log8 4 
2 1
19. log7  2 20. log6 64 4
3 49

Include the following in your answer: 21. log3  1

1
3
22. log4  4
1
256
23. log9 9(n  1) n  1 24. 2log2 32 32

• the relative intensities of a pin drop, a whisper, normal conversation, kitchen Solve each equation or inequality. Check your solutions.
noise, and a jet engine written in scientific notation, 25. log10 n
3 
1
1000
26. log4 x 3 x  64 27. log4 x
 8
3
2

28. log15 x
3 125 29. log7 q  0 0  q  1 30. log6 (2y  8)  2 y  14
• a plot of each of these relative intensities on the scale shown below, and 1 1
31. logy 16
4  32. logn 
3 2 33. logb 1024
5 4
2 8

34. log8 (3x  7)  log8 (7x  4) 35. log7 (8x  20)

log7 (x  6) 36. log3 (x2  2)
log3 x
11 11 11 11 12
0 2  10 4  10 6  10 8  10 1  10 x
3
4
2 2

37. SOUND Sounds that reach levels of 130 decibels or more are painful to humans. What
• an explanation as to why the logarithmic scale might be preferred over the is the relative intensity of 130 decibels? 1013

scale shown above. 38. INVESTING Maria invests $1000 in a savings account that pays 8% interest
compounded annually. The value of the account A at the end of five years can be
determined from the equation log A
log[1000(1  0.08)5]. Find the value of A to the
nearest dollar. $1469

Standardized 74. What is the equation of the function graphed Gl

Reading
NAME
/M G ______________________________________________
Hill 582 DATE ____________
Gl PERIOD
Al _____
b 2

Readingto
to Learn
y
10-2 Learn Mathematics
Test Practice at the right? B
(1, 6) Mathematics, p. 583
Logarithms and Logarithmic Functions
ELL
A y
2(3)x Pre-Activity Why is a logarithmic scale used to measure sound?
Read the introduction to Lesson 10-2 at the top of page 531 in your textbook.
x
y
2
How many times louder than a whisper is normal conversation?
B
1 104 or 10,000 times
3 (0, 2) Reading the Lesson
1 x 1. a. Write an exponential equation that is equivalent to log3 81
4. 34  81
C  
y
3 
2 O x
b. Write a logarithmic equation that is equivalent to 25

c. Write an exponential equation that is equivalent to log4 1

0. 40  1
12 1

 . log25    
5 5 2
1 1

d. Write a logarithmic equation that is equivalent to 103

0.001. log10 0.001  3
D y
3(2)x e. What is the inverse of the function y
5x? y  log5 x
f. What is the inverse of the function y
log10 x? y  10x
www.algebra2.com/self_check_quiz Lesson 10-2 Logarithms and Logarithmic Functions 537

Lesson 10-2
2. Match each function with its graph.
 13 
x
a. y
3x IV b. y
log3 x I c. y
 II
I. y II. y III. y

NAME ______________________________________________ DATE ____________ PERIOD _____ O x

Enrichment,
10-2 Enrichment p. 584 O x O x

Musical Relationships 3. Indicate whether each of the following statements about the exponential function
The frequencies of notes in a musical scale that are one octave apart are y
log5 x is true or false.
related by an exponential equation. For the eight C notes on a piano, the a. The y-axis is an asymptote of the graph. true
equation is Cn
C12n  1, where Cn represents the frequency of note Cn.
b. The domain is the set of all real numbers. false
c. The graph contains the point (5, 0). false
d. The range is the set of all real numbers. true
e. The y-intercept is 1. false

Helping You Remember

1. Find the relationship between C1 and C2. C2  2C1 4. An important skill needed for working with logarithms is changing an equation between
logarithmic and exponential forms. Using the words base, exponent, and logarithm, describe
2. Find the relationship between C1 and C4. C4  8C1 an easy way to remember and apply the part of the definition of logarithm that says,
“logb x
y if and only if b y
x.” Sample answer: In these equations, b stands
The frequencies of consecutive notes are related by a for base. In log form, b is the subscript, and in exponential form, b is the
common ratio r. The general equation is fn
f1r n  1. number that is raised to a power. A logarithm is an exponent, so y, which is
the log in the first equation, becomes the exponent in the second equation.
3. If the frequency of middle C is 261.6 cycles per second
and the frequency of the next higher C is 523.2 cycles

Lesson 10-2 Logarithms and Logarithmic Functions 537

 2
75. In the figure at the right, if y
x and z
3w,

4 Assess then x
D
A 14. B
7

20. x˚ y˚
z˚
C 28. D 35.
Open-Ended Assessment w˚

Writing Have students write a

step-by-step explanation of the
procedure for solving a logarith- Maintain Your Skills
7
mic equation such as log8 n
. Mixed Review Simplify each expression. (Lesson 10-1)
3
76. x6  x6 x26 77. b624
 b12
Getting Ready for 14
Solve each equation. Check your solutions. (Lesson 9-6) 79. 3, 
Lesson 10-3 2x  1 x1 20 2a  5 a3
5
5
78.   
 2  ⵰
x4 x  4x
79.   

a9 2 
3a  2 3a  25a  18
PREREQUISITE SKILL In Lesson x
10-3, students will evaluate ex- Solve each equation by using the method of your choice. Find exact solutions.
pressions using the properties of (Lesson 6-5)
7 
5 73
logarithms. Because these proper- 80. 9y2
49  81. 2p2
5p  6 
3 4
ties are related to exponential Simplify each expression. (Lesson 9-2) 83. 
6x  58
properties, students must be (x  3)(x  3)(x  7)
3 4 7 43 x7 x3
familiar with exponential proper- 82.       83.   
2y 3y 5y 30y x2  9 x2  10x  21
ties when multiplying or divid-
ing terms with like bases. Use 84. BANKING Donna Bowers has $4000 she wants to save in the bank. A certificate
of deposit (CD) earns 8% annual interest, while a regular savings account earns
Exercises 85–90 to determine your 3% annual interest. Ms. Bowers doesn’t want to tie up all her money in a CD,
students’ familiarity with multi- but she has decided she wants to earn $240 in interest for the year. How much
plying and dividing monomials. money should she put in to each type of account? (Hint: Use Cramer’s Rule.)
(Lesson 4-4) $2400, CD; $1600, savings
Assessment Options
Getting Ready for PREREQUISITE SKILL Simplify. Assume that no variable equals zero.
Practice Quiz 1 The quiz the Next Lesson (To review multiplying and dividing monomials, see Lesson 5-1.)
provides students with a brief 85. x4  x6 x10 86. (y3)8 y24 87. (2a2b)3 8a6b3
review of the concepts and skills x5yz2 x3 7 0
90. 4  1
a4n7 b
in Lessons 10-1 and 10-2. Lesson 88. 3 an6 89.   
an x2y3z5 y2z3 a
numbers are given to the right of
the exercises or instruction lines
so students can review concepts
P ractice Quiz 1 Lessons 10-1 and 10-2
not yet mastered. 1. Determine whether 5(1.2)x represents exponential growth or decay. (Lesson 10-1) growth
Quiz (Lessons 10-1 and 10-2) is 2. Write an exponential function whose graph passes through (0, 2) and (2, 32). y  2(4)x
available on p. 623 of the
Chapter 10 Resource Masters. 3. Write an equivalent logarithmic equation for 46
4096. (Lesson 10-2) log4 4096  6
3 3
4. Write an equivalent exponential equation for log9 27
. (Lesson 10-2) 
2 9 2  27
Evaluate each expression. (Lesson 10-2)
4
5. log8 16  6. log4 415 15
3
Solve each equation or inequality. Check your solutions. (Lessons 10-1 and 10-2)
3 1
7. 34x
33  x  8. 32n   n
1
5 9
9. log2 (x  6) 5 x  26 10. log5 (4x  1)
log5 (3x  2) 3

538 Chapter 10 Exponential and Logarithmic Relations

538 Chapter 10 Exponential and Logarithmic Relations

 Graphing
Calculator
A Follow-Up of Lesson 10-2
Investigation
A Follow-Up of Lesson 10-2
Modeling Real-World Data: Curve Fitting Getting Started
We are often confronted with data for which we need to find an equation that best
fits the information. We can find exponential and logarithmic functions of best fit Turning Off Stat Plots Before
using a TI-83 Plus graphing calculator. Step 1, students should use the
keystrokes 2nd [STAT PLOT] and
Example check that both plot 2 and plot 3
The population per square mile in the United are turned off.
U.S. Population Density
States has changed dramatically over a period Diagnostics Display Students
of years. The table shows the number of people People per People per
per square mile for several years.
Year
square mile
Year
square mile should have the calculator set to
1790 4.5 1900 21.5 DiagnosticOn. To set the calculator
a. Use a graphing calculator to enter the data
and draw a scatter plot that shows how the 1800 6.1 1910 26.0 for diagnostics, use 2nd [CATALOG],
number of people per square mile is related 1810 4.3 1920 29.9 move the cursor down to
to the year.
1820 5.5 1930 34.7 DiagnosticOn, and press ENTER
Step 1 Enter the year into L1 and the people per 1830 7.4 1940 37.2 twice.
square mile into L2.
1840 9.8 1950 42.6
KEYSTROKES: See pages 87 and 88 to review
1850 7.9 1960 50.6
how to enter lists.
Be sure to clear the Y list. Use the
1860 10.6 1970 57.5 Teach
1870 10.9 1980 64.0
key to move the cursor from L1 to L2. • When students begin the exer-
1880 14.2 1990 70.3
Step 2 Draw the scatter plot. 1890 17.8 2000 80.0 cises, they should clear lists L1
KEYSTROKES: See pages 87 and 88 to review Source: Northeast-Midwest Institute
and L2. They should also enter
how to graph a scatter plot. appropriate settings for the
Make sure that Plot 1 is on, the scatter plot is chosen, Xlist is L1, and Ylist is L2.
graphing window.
Use the viewing window [1780, 2020] with a scale factor of 10 by [0, 115] • Point out that the table of data
with a scale factor of 5. is arranged in two “double”
We see from the graph that the equation that
columns.
best fits the data is a curve. Based on the • Suggest that students compare
shape of the curve, try an exponential model. their graphs to the one shown.
• Have students estimate the
population density in 2010 and
2050. How soon will the popu-
[1780, 2020] scl: 10 by [0, 115] scl: 5 lation density be twice what it
was in 2000? about 2040
Step 3 To determine the exponential equation that best fits the data, use the
exponential regression feature of the calculator. • If you have time, consider
extending this activity into a
KEYSTROKES: STAT 0 2nd [L1] , 2nd [L2] ENTER discussion of how life in the
The equation is y
1.835122  1011(1.014700091)x. future will be different as the
(continued on the next page) result of the increasing popula-
tion density. Ask students to
www.algebra2.com/other_calculator_keystrokes
think about the effect on trans-
Graphing Calculator Investigation Modeling Real-World Data: Curve Fitting 539 portation, housing, crime rates,
and so on. You may wish to
team-teach with a social
studies teacher.

Graphing Calculator Investigation Modeling Real-World Data: Curve Fitting 539

 Graphing Calculator Investigation

The calculator also reports an r value of 0.991887235. Recall that this number is a
Assess correlation coefficient that indicates how well the equation fits the data. A perfect fit
would be r
1. Therefore, we can conclude that this equation is a pretty good fit for
the data.
In Exercise 2, make sure students To check this equation visually, overlap the graph of the equation with the scatter plot.
can explain why their equation
KEYSTROKES: VARS 5 1 GRAPH
of best fit is a good choice. In
Exercise 3, students’ answers may
vary slightly. When you discuss
Exercise 6, you may want to ask
for any ideas students have
about how to use the calculator [1780, 2020] scl: 10 by [0, 115] scl: 5
to judge the relative merits of
various models (quadratic, cubic, b. If this trend continues, what will be the population per square mile in 2010?
quartic, and exponential). To determine the population per square mile in 2010, from the graphics screen, find
the value of y when x
2010.
KEYSTROKES: 2nd [CALC] 1 2010 ENTER
Answers
1.

[1780, 2020] scl: 10 by [0, 115] scl: 5

The calculator returns a value of approximately 100.6. If this trend continues, in 2010,
there will be approximately 100.6 people per square mile.
[0, 50] scl: 5 by [30, 400] scl: 20
Exercises
2.
In 1985, Erika received $30 from her aunt and uncle for Elapsed
her seventh birthday. Her father deposited it into a bank Balance
Time (years)
account for her. Both Erika and her father forgot about
0 $30.00
the money and made no further deposits or withdrawals.
The table shows the account balance for several years. 5 $41.10
1. Use a graphing calculator to draw a scatter plot for the data. 10 $56.31
See margin. 15 $77.16
2. Calculate and graph the curve of best fit that shows how
[0, 50] scl: 5 by [30, 400] scl: 20 20 $105.71
the elapsed time is related to the balance. Use ExpReg for
this exercise. See margin. 25 $144.83
3. Write the equation of best fit. y  29.99908551(1.06500135)x 30 $198.43

4. Write a sentence that describes the fit of the graph to the data.
This equation is a good fit because r 1.
5. Based on the graph, estimate the balance in 41 years. Check
this using the CALC value. After 41 years she will have approximately $397.
6. Do you think there are any other types of equations that would be good models
for these data? Why or why not? A quadratic equation might be a good model
for this example because the shape is close to a portion of a parabola.

540 Chapter 10 Exponential and Logarithmic Relations

540 Chapter 10 Exponential and Logarithmic Relations

Properties of Logarithms Lesson
Notes

• Simplify and evaluate expressions using the properties of logarithms.

• Solve logarithmic equations using the properties of logarithms.
1 Focus
are the properties of exponents and logarithms related?
In Lesson 5-1, you learned that the product of powers is the sum of their
5-Minute Check
exponents. Transparency 10-3 Use as
9  81
32  34 or 32  4 a quiz or review of Lesson 10-2.
In Lesson 10-2, you learned that logarithms are exponents, so you might expect
that a similar property applies to logarithms. Let’s consider a specific case. Does
Mathematical Background notes
log3 (9  81)
log3 9  log3 81? are available for this lesson on
p. 520D.
log3 (9  81)
log3 (32  34) Replace 9 with 32 and 81 with 34.

log3 3(2  4) Product of Powers are the properties of

2  4 or 6 Inverse property of exponents and logarithms exponents and
logarithms related?
log3 9  log3 81
log3 32  log3 34 Replace 9 with 32 and 81 with 34.

2  4 or 6 Inverse property of exponents and logarithms
Ask students:
• How do you know that
So, log3 (9  81)
log3 9  log3 81.
logarithms are exponents?
Sample answer: The logarithm of a
number is equal to the power (or
PROPERTIES OF LOGARITHMS Since logarithms are exponents, the exponent) when the number is
properties of logarithms can be derived from the properties of exponents. The rewritten in exponential form.
example above and other similar examples suggest the following property of
logarithms. • Since log3 (9  81)
log3 729,
how could log3 729 have been
Product Property of Logarithms used in the justification that
log3 (9  81)
log3 9  log3 81?
• Words The logarithm of a product is the sum of the logarithms of its factors.
After stating that log3 (9  81) 
• Symbols For all positive numbers m, n, and b, where b  1,
logb mn
logb m  logb n. log3 729, then the statements
• Example log3 (4)(7)
log3 4  log3 7
log3 729  log3 36 and log3 36  6
could be used to justify that
log3 (9  81)  6.
To show that this property is true, let bx
m and by
n. Then, using the definition
of logarithm, x
logb m and y
logb n.
bxby
mn
bx  y
mn Product of Powers

logb bx  y
logb mn Property of Equality for Logarithmic Functions

x  y
logb mn Inverse Property of Exponents and Logarithms
logb m  logb n
logb mn Replace x with logb m and y with logb n.

You can use the Product Property of Logarithms to approximate logarithmic

expressions.
Lesson 10-3 Properties of Logarithms 541

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters 5-Minute Check Transparency 10-3
• Study Guide and Intervention, pp. 585–586 Answer Key Transparencies
• Skills Practice, p. 587
• Practice, p. 588 Technology
• Reading to Learn Mathematics, p. 589 Interactive Chalkboard
• Enrichment, p. 590
• Assessment, pp. 623, 625

Lesson x-x Lesson Title 541

 Example 1 Use the Product Property
2 Teach Use log2 3  1.5850 to approximate the value of log2 48.
log2 48
log2 (24  3) Replace 48 with 16  3 or 24  3.
PROPERTIES OF TEACHING TIP
The value of R in

log2 24  log2 3 Product Property
LOGARITHMS Example 3 is determined
4  log2 3 Inverse Property of Exponents and Logarithms
by finding the ratio of the
In-Class Examples Power
Point® intensity l of the sound in  4  1.5850 or 5.5850 Replace log2 3 with 1.5850.
watts per square meter to Thus, log2 48 is approximately 5.5850.
Teaching Tip When discussing the intensity l0 of 1012
watts per square meter.
the Product Property of The intensity l0 corre- Recall that the quotient of powers is found by subtracting exponents. The
Logarithms, point out that the sponds to the threshold property for the logarithm of a quotient is similar.
logarithms used in the example of hearing. Thus a formu-
(log3 (4)(7)
log3 4  log37) la that relates the
Quotient Property of Logarithms
show the property applies to all intensity of a sound in
watts per square meter to • Words The logarithm of a quotient is the difference of the logarithms of the
logarithms and not just those its loudness in decibels numerator and the denominator.
that can be simplified. Be sure is L  10 log10  l.
• Symbols For all positive numbers m, n, and b, where b  1,
students did not get this impres- l0 m
logb 
logb m  logb n.
sion from the earlier example n
where it was shown that More About . . .
log3 (9  81)
log3 9  log3 81. You will show that this property is true in Exercise 47.

1 Use log5 2  0.4307 to approx- Example 2 Use the Quotient Property

imate the value of log5 250.
Use log3 5  1.4650 and log3 20  2.7268 to approximate log3 4.
3.4307
20 20
log3 4
log3  Replace 4 with the quotient .
Teaching Tip Some students 5 5

may wonder how the approxima-

log3 20  log3 5 Quotient Property
tion for log2 3 was determined  2.7268  1.4650 or 1.2618 log3 20
2.7268 and log3 5
1.4650
since on most calculators the Sound Technician
Thus, log3 4 is approximately 1.2618.
log button calculates only Sound technicians produce
movie sound tracks in
logarithms of base 10. State that CHECK Using the definition of logarithm and a calculator, 31.2618  4. ⻫
motion picture production
log 3
10
log2 3
 , which can be studios, control the sound
log10 2 of live events such as
evaluated using a calculator. concerts, or record music Example 31 Use Properties of Logarithms
in a recording studio.
Stress that this procedure will SOUND The loudness L of a sound in decibels is given by L  10 log10 R, where
be formally discussed in Online Research R is the sound’s relative intensity. Suppose one person talks with a relative
Lesson 10-4. For information about intensity of 106 or 60 decibels. Would the sound of ten people each talking at
a career as a sound that same intensity be ten times as loud or 600 decibels? Explain your reasoning.
2 Use log6 8  1.1606 and technician, visit:
Let L1 be the loudness of one person talking. → L1
10 log10 106
www.algebra2.com/
log6 32  1.9343 to approxi- Let L2 be the loudness of ten people talking. → L2
10 log10 (10  106)
careers
mate the value of log6 4. Then the increase in loudness is L2  L1.
0.7737
L2  L1
10 log10 (10  106)  10 log10 106 Substitute for L1 and L2.

3 SOUND The sound made by

10(log10 10  log10 106)  10 log10 106 Product Property
a lawnmower has a relative
10 log10 10  10 log10 106  10 log10 106 Distributive Property
intensity of 109 or 90 decibels.
Would the sound of ten
10 log10 10 Subtract.

lawnmowers running at that

10(1) or 10 Inverse Property of Exponents and Logarithms
same intensity be ten times The sound of two people talking is perceived by the human ear to be only about
as loud or 900 decibels? 10 decibels louder than the sound of one person talking, or 70 decibels.
Explain your reasoning. No;
the sound of ten lawnmowers is 542 Chapter 10 Exponential and Logarithmic Relations
perceived to be only 10 decibels
louder than the sound of one
lawnmower, or 100 decibels.
Unlocking Misconceptions
Power Property After you have discussed the Power Property of
Logarithms on p. 543, clarify that the property works for logarithms
because they are equivalent to exponents. Stress that students should
not read a statement such as log2 53
3 log2 5 and conclude that
53
3 5.

542 Chapter 10 Exponential and Logarithmic Relations

 Recall that the power of a power is found by multiplying exponents. The property
for the logarithm of a power is similar. In-Class Example Power
Point®

Power Property of Logarithms 4 Given log5 6  1.1133,

• Words The logarithm of a power is the product of the logarithm and the approximate the value of
exponent. log5 216. 3.3399
• Symbols For any real number p and positive numbers m and b, where b  1,
logb mp
p logb m.

You will show that this property is true in Exercise 50. SOLVE LOGARITHMIC
Example 4 Power Property of Logarithms EQUATIONS
Given log4 6  1.2925, approximate the value of log4 36. In-Class Example Power
Point®
log4 36
log4 62 Replace 36 with 62.

2 log4 6 Power Property 5 Solve each equation.
 2(1.2925) or 2.585 Replace log4 6 with 1.2925. a. 4 log2 x  log2 5
log2 125 5
b. log8 x  log8 (x  12)
2 16
SOLVE LOGARITHMIC EQUATIONS You can use the properties of
logarithms to solve equations involving logarithms.

Example 5 Solve Equations Using Properties of Logarithms

Solve each equation.
a. 3 log5 x  log5 4  log5 16
3 log5 x  log5 4
log5 16 Original equation
log5 x3  log5 4
log5 16 Power Property
x3
log5 
log5 16 Quotient Property
4
x3

16 Property of Equality for Logarithmic Functions
4
x3
64 Multiply each side by 4.
x
4 Take the cube root of each side.
The solution is 4.

b. log4 x  log4 (x  6)  2
log4 x  log4 (x  6)
2 Original equation
log4 x(x  6)
2 Product Property
x(x  6)
42 Definition of logarithm
x  6x  16
0
2 Subtract 16 from each side.
(x  8)(x  2)
0 Factor.
x  8
0 or x  2
0 Zero Product Property
Study Tip x
8 x
2 Solve each equation.
Checking CHECK Substitute each value into the original equation.
Solutions
It is wise to check all log4 8  log4 (8  6)  2 log4 (2)  log4 (2  6)  2
solutions to see if they are log4 8  log4 2  2 log4 (2)  log4 (8)  2
valid since the domain of
a logarithmic function is log4 (8  2)  2 Since log4 (2) and log4 (8) are
not the complete set of log4 16  2 undefined, 2 is an extraneous
real numbers. 2
2 ⻫ solution and must be eliminated.
The only solution is 8.

www.algebra2.com/extra_examples Lesson 10-3 Properties of Logarithms 543

Differentiated Instruction
Interpersonal Right after discussing Example 5, have pairs of students
rework both parts of the example together without looking at the
solution in the text. Have the partners take turns explaining the solution
steps to each other.

Lesson 10-3 Properties of Logarithms 543

 3 Practice/Apply Concept Check 1. Name the properties that are used to derive the properties of logarithms.
1. properties of expo-
2. OPEN ENDED Write an expression that can be simplified by using two or more
nents
properties of logarithms. Then simplify it.
2. Sample answer:
Study Notebook 2 log3 x  log3 5; 3. FIND THE ERROR Umeko and Clemente are simplifying log7 6  log7 3  log7 2.

Have students— log3 5x2

Umeko Clemente
• add the definitions/examples of
log7 6 + log7 3 - log7 2 log 7 6 + log 7 3 - log 7 2
the vocabulary terms to their
Vocabulary Builder worksheets for = log7 18 - log7 2 = log 7 9 - log 7 2
Chapter 10. = log7 9 = log 7 7 or 1
• summarize the properties of loga-
rithms they learned in this lesson. Who is correct? Explain your reasoning. Umeko; see margin for explanation.
• include any other item(s) that they
find helpful in mastering the skills Guided Practice Use log3 2  0.6310 and log3 7  1.7712 to approximate the value of each expression.
7 2
GUIDED PRACTICE KEY 4. log3  1.1402 5. log3 18 2.6310 6. log3  0.3690
in this lesson. 2 3
Exercises Examples
Solve each equation. Check your solutions.
4–6 1, 2, 4
7–10 5 7. log3 42  log3 n
log3 7 6 8. log2 3x  log2 5
log2 30 2
11, 12 3
9. 2 log5 x
log5 9 3 10. log10 a  log10 (a  21)
2 4

FIND THE ERROR Application MEDICINE For Exercises 11 and 12, use the following information.
When discussing The pH of a person’s blood is given by pH
6.1  log10 B  log10 C, where B is the
the error made by concentration of bicarbonate, which is a base, in the blood and C is the concentration
of carbonic acid in the blood. 11. pH  6.1  log B
Clemente, remind students that 10 C
11. Use the Quotient Property of Logarithms to simplify the formula for blood pH.
logarithms are exponents. Add-
ing log7 6  log7 3 as log7 (6  3) 12. Most people have a blood pH of 7.4. What is the approximate ratio of
is similar to saying that bicarbonate to carbonic acid for blood with this pH? 20:1
x2  x3
x2  3 or x5, which ★ indicates increased difficulty
students should recognize as
being untrue because x2 and x3 Practice and Apply
are unlike terms. Homework Help Use log5 2  0.4307 and log5 3  0.6826 to approximate the value of each
For See expression.
Exercises Examples 2 3
13–20 1, 2, 4 13. log5 9 1.3652 14. log5 8 1.2921 15. log5  0.2519 16. log5  0.2519
3 2
About the Exercises… 21–34
37–45
5
3 17. log5 50 2.4307 18. log5 30 ★ 19. log5 0.5
10
★ 20. log5 9 0.0655
Organization by Objective 2.1133 0.4307
• Properties of Logarithms: Extra Practice Solve each equation. Check your solutions.
See page 850.
13–20, 37–46 21. log3 5  log3 x
log3 10 2 22. log4 a  log4 9
log4 27 3
• Solve Logarithmic 23. log10 16  log10 2t
log10 2 4 24. log7 24  log7 (y  5)
log7 8 2
Equations: 21–34 1 1 1
25. log2 n
 log2 16   log2 49 14 26. 2 log10 6   log10 27
log10 x 12
Odd/Even Assignments 4 2 3
Exercises 13–34 are structured 27. log10 z  log10 (z  3)
1 2 28. log6 (a2  2)  log6 2
2 4
so that students practice the 29. log2 (12b  21)  log2 (b2  3)
2  30. log2 (y  2)  log2 (y  2)
1 6
same concepts whether they are 8
31. log3 0.1  2 log3 x
log3 2  log3 5 10 32. log5 64  log5   log5 2
log5 4p 12
assigned odd or even problems. 3
544 Chapter 10 Exponential and Logarithmic Relations
Alert! Exercise 46 involves
research on the Internet or
other reference materials.
Answer
Assignment Guide 3. Clemente incorrectly applied the product and quotient properties of logarithms.
Basic: 13–17 odd, 21–31 odd, log7 6  log7 3  log7 (6  3) or log7 18 Product Property of Logarithms
35–40, 47–66
log7 18  log7 2  log7 (18  2) or log7 9 Quotient Property of Logarithms
Average: 13–33 odd, 35–43,
47–66
Advanced: 14–34 even, 35, 36,
41–62 (optional: 63–66)

544 Chapter 10 Exponential and Logarithmic Relations

 1
35. False; Solve for n. 34. 2(x  1) Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

3 10-3 Study Guide and

log2 (2  2 ) 
2 3
★ 33. loga 4n  2 loga x
loga x x ★ 34. logb 8  3 logb n
3 logb (x  1) p. 585 (shown)
Properties and p. 586
of Logarithms
4
log2 12, Properties of Logarithms Properties of exponents can be used to develop the
following properties of logarithms.

log2 22  log2 23  CRITICAL THINKING Tell whether each statement is true or false. If true, show Product Property
of Logarithms
For all positive numbers m, n, and b, where b  1,
logb mn
logb m  logb n.

2  3 or 5, that it is true. If false, give a counterexample. Quotient Property

of Logarithms
For all positive numbers m, n, and b, where b  1,
logb m

logb m  logb n.
n

and log2 12  5 since 35. For all positive numbers m, n, and b, where b  1, logb (m  n)
logb m  logb n. Power Property
of Logarithms
For any real number p and positive numbers m and b,
where b  1, logb m p
p logb m.

25  12. 36. For all positive numbers m, n, x, and b, where b  1, n logb x  m logb x
Example Use log3 28  3.0331 and log3 4  1.2619 to approximate

(n  m) logb x. See pp. 573A–573D. the value of each expression.

a. log3 36 b. log3 7 c. log3 256
log3 36

log3 (32  4)
log3 32  log3 4
log3 7
log3  284  log3 256

log3 (44)
4  log3 4

log3 28  log3 4
37. EARTHQUAKES The great Alaskan earthquake in 1964 was about 100 times
2  log3 4
 3.0331  1.2619
 4(1.2619)
 2  1.2619  5.0476
more intense than the Loma Prieta earthquake in San Francisco in 1989. Find  3.2619
 1.7712

the difference in the Richter scale magnitudes of the earthquakes. 2 Exercises

39. about 0.4214 Use log12 3  0.4421 and log12 7  0.7831 to evaluate each expression.

Lesson 10-3
kilocalorie per gram BIOLOGY For Exercises 3840, use the following information. 1. log12 21 1.2252
7
2. log12  0.3410
3
3. log12 49 1.5662

The energy E (in kilocalories per gram molecule) needed to transport a substance 27
40. about 0.8429 4. log12 36 1.4421 5. log12 63 1.6673 6. log12  0.2399
49
from the outside to the inside of a living cell is given by E
1.4(log10 C2  log10 C1),
kilocalories per gram 81
7. log12  0.2022 8. log12 16,807 3.9155 9. log12 441 2.4504
where C1 is the concentration of the substance outside the cell and C2 is the 49

concentration inside the cell. C2

38. Express the value of E as one logarithm. E  1.4 log C
Use log5 3  0.6826 and log5 4  0.8614 to evaluate each expression.

10. log5 12 1.5440 11. log5 100 2.8614 12. log5 0.75 0.1788
1
39. Suppose the concentration of a substance inside the cell is twice the 13. log5 144 3.0880
27
14. log5  0.3250 15. log5 375 3.6826
16
concentration outside the cell. How much energy is needed to transport the
substance on the outside of the cell to the inside? (Use log10 2  0.3010.)
9 81
 0.1788
16. log5 1.3 17. log5  0.3576 18. log5  1.7304
16 5

40. Suppose the concentration of a substance inside the cell is four times the Gl NAME
/M G ______________________________________________
Hill 585 DATE ____________
GlPERIOD
Al _____
b 2

concentration outside the cell. How much energy is needed to transport the Skills
10-3 Practice,
Practice (Average)
p. 587 and
substance from the outside of the cell to the inside? Practice,
Properties p. 588 (shown)
of Logarithms
Use log10 5  0.6990 and log10 7  0.8451 to approximate the value of each
expression.
7 5
1. log10 35 1.5441 2. log10 25 1.3980 3. log10  0.1461 4. log10  0.1461
SOUND For Exercises 41–43, use the formula for the loudness of sound in 5 7

Example 3 on page 542. Use log10 2  0.3010 and log10 3  0.47712. 5. log10 245 2.3892 6. log10 175 2.2431
25
7. log10 0.2 0.6990 8. log10  0.5529

Star Light 7

The Greek astronomer

41. A certain sound has a relative intensity of R. By how many decibels does the Solve each equation. Check your solutions.
2 3

Hipparchus made the first sound increase when the intensity is doubled? 3 9. log7 n
 log7 8 4
3
10. log10 u
 log10 4 8
2

11. log6 x  log6 9

log6 54 6 12. log8 48  log8 w
log8 4 12
known catalog of stars. He 42. A certain sound has a relative intensity of R. By how many decibels does the
listed the brightness of sound decrease when the intensity is halved? 3 13. log9 (3u  14)  log9 5
log9 2u 2 14. 4 log2 x  log2 5
log2 405 3

each star on a scale of 1 to 1

★ 43. A stadium containing 10,000 cheering people can produce a crowd noise of
1
15. log3 y
log3 16   log3 64  16. log2 d
5 log2 2  log2 8 4
3 4
6, the brightest being 1. 17. log10 (3m  5)  log10 m
log10 2 2 18. log10 (b  3)  log10 b
log10 4 1
With no telescope, he about 90 decibels. If every one cheers with the same relative intensity, how
could only see stars as dim much noise, in decibels, is a crowd of 30,000 people capable of producing? 19. log8 (t  10)  log8 (t  1)
log8 12 2 20. log3 (a  3)  log3 (a  2)
log3 6 0

as the 6th magnitude. Explain your reasoning. About 95 decibels; see margin for explanation. 21. log10 (r  4)  log10 r
log10 (r  1) 2 22. log4 (x2  4)  log4 (x  2)
log4 1 3

Source: NASA 23. log10 4  log10 w

2 25 24. log8 (n  3)  log8 (n  4)
1 4

STAR LIGHT For Exercises 44–46, use the 25. 3 log5 (x2  9)  6
0 4 26. log16 (9x  5)  log16 (x2  1)
 3
1
2

following information. 27. log6 (2x  5)  1

log6 (7x  10) 8 28. log2 (5y  2)  1
log2 (1  2y) 0

The brightness, or apparent magnitude, 29. log10 (c2  1)  2

log10 (c  1) 101 30. log7 x  2 log7 x  log7 3
log7 72 6
m of a star or planet is given by the formula
L Moon Sirius 31. SOUND The loudness L of a sound in decibels is given by L
10 log10 R, where R is the

m
6  2.5 log10 , where L is the amount sound’s relative intensity. If the intensity of a certain sound is tripled, by how many
decibels does the sound increase? about 4.8 db
L0 The crescent moon is about 100 times
of light coming to Earth from the star or brighter than the brightest star, Sirius. 32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,
and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude
reading m is given by m
log10 x, where x represents the amplitude of the seismic wave
planet and L0 is the amount of light from a causing ground motion. How many times greater is the amplitude of an earthquake that
measures 4.5 on the Richter scale than one that measures 3.5? 10 times
sixth magnitude star.
NAME ______________________________________________ DATE ____________
Gl PERIOD
Al _____

★ 44. Find the difference in the magnitudes

Gl /M G Hill 588 b 2
Reading
10-3 Readingto
to Learn
Learn Mathematics
of Sirius and the crescent moon. 5 Mathematics, p. 589 ELL
Properties of Logarithms

★ 45. Find the difference in the magnitudes

Pre-Activity How are the properties of exponents and logarithms related?
Read the introduction to Lesson 10-3 at the top of page 541 in your textbook.

of Saturn and Neptune. 7.5 Find the value of log5 125. 3 Find the value of log5 5. 1
Find the value of log5 (125  5). 2
Which of the following statements is true? B
46. RESEARCH Use the Internet or other Saturn Neptune A. log5 (125  5)
(log5 125)  (log5 5)
B. log5 (125  5)
log5 125  log5 5
reference to find the magnitude of the Saturn, as seen from Earth, is
1000 times brighter than Neptune.
dimmest stars that we can now see with Reading the Lesson
1. Each of the properties of logarithms can be stated in words or in symbols. Complete the
ground-based telescopes. about 22 statements of these properties in words.
a. The logarithm of a quotient is the difference of the logarithms of the
numerator denominator .
www.algebra2.com/self_check_quiz Lesson 10-3 Properties of Logarithms 545
and the
b. The logarithm of a power is the product of the logarithm of the base and
the exponent .
c. The logarithm of a product is the sum of the logarithms of its
factors .

NAME ______________________________________________ DATE ____________ PERIOD _____ 2. State whether each of the following equations is true or false. If the statement is true,
Answer name the property of logarithms that is illustrated.
Enrichment, p. 590 Lesson 10-3
10-3 Enrichment a. log3 10
log3 30  log3 3 true; Quotient Property
b. log4 12
log4 4  log4 8 false
43. L  10 log10 R, where L is the loudness of Spirals c. log2 81
2 log2 9 true; Power Property
d. log8 30
log8 5  log8 6 false
the sound in decibels and R is the relative Consider an angle in standard position with its vertex at a point O called the
pole. Its initial side is on a coordinatized axis called the polar axis. A point P 3. The algebraic process of solving the equation log2 x  log2 (x  2)
3 leads to “x
4
on the terminal side of the angle is named by the polar coordinates (r, ), or x
2.” Does this mean that both 4 and 2 are solutions of the logarithmic equation?
intensity of the sound. Since the crowd where r is the directed distance of the point from O and  is the measure of
the angle. Graphs in this system may be drawn on polar coordinate paper
Explain your reasoning. Sample answer: No; 2 is a solution because it
checks: log2 2  log2 (2  2)  log2 2  log2 4  1  2  3. However,
such as the kind shown below.
increased by a factor of 3, we assume that 100 90 80
because log2 (4) and log2 ( 2) are undefined, 4 is an extraneous
solution and must be eliminated. The only solution is 2.
110 70

the intensity also increases by a factor of 3. 130

120 60
50
Helping You Remember
140 40

Thus, we need to find the loudness of 3R. 150 30

4. A good way to remember something is to relate it something you already know. Use words
to explain how the Product Property for exponents can help you remember the product
property for logarithms. Sample answer: When you multiply two numbers or
L  10 log10 3R; L  10(log10 3  log10 R) 160 20
expressions with the same base, you add the exponents and keep the
170 10 same base. Logarithms are exponents, so to find the logarithm of a

L  10 log10 3  10 log10 R;
product, you add the logarithms of the factors, keeping the same base.
180 0

L  10(0.4771)  90; L  4.771  90 or about 95

190 350

Lesson 10-3 Properties of Logarithms 545

 47. CRITICAL THINKING Use the properties of exponents to prove the Quotient

4 Assess Property of Logarithms. See margin.

48. WRITING IN MATH Answer the question that was posed at the beginning of
Open-Ended Assessment the lesson. See pp. 573A–573D.
How are the properties of exponents and logarithms related?
Speaking Ask students to explain
the Product Property, Quotient Include the following in your answer:
Property, and Power Property of • examples like the one shown at the beginning of the lesson illustrating the
Quotient Property and Power Property of Logarithms, and
Logarithms in their own words.
• an explanation of the similarity between one property of exponents and its
Encourage them to use specific related property of logarithms.
examples for clarification.
Standardized 49. Simplify 2 log5 12  log5 8  2 log5 3. A
Getting Ready for Test Practice A log5 2 B log5 3 C log5 0.5 D 1

Lesson 10-4 50. SHORT RESPONSE Show that logb mp

p logb m for any real number p and
PREREQUISITE SKILL Students positive number m and b, where b  1. See margin.
will use common logarithms to
solve exponential equations and Maintain Your Skills
inequalities in Lesson 10-4. The
solution techniques involve using Mixed Review Evaluate each expression. (Lesson 10-2)
the skills they learned when 1
51. log3 81 4 52. log9  3 53. log7 72x 2x
729
solving logarithmic equations and
inequalities. Use Exercises 63–66 Solve each equation or inequality. Check your solutions. (Lesson 10-1)
to determine your students’ 54. 35n  3
333 6 55. 7a
494 8 56. 3d  4 9d d  4
familiarity with solving logarith-
Determine whether each graph represents an odd-degree polynomial function
mic equations and inequalities. or an even-degree polynomial function. Then state how many real zeros each
function has. (Lesson 7-1)
Assessment Options 57. y odd; 3 58. y even; 4
Quiz (Lesson 10-3) is available
on p. 623 of the Chapter 10
Resource Masters. O x O x
Mid-Chapter Test (Lessons 10-1
through 10-3) is available on
p. 625 of the Chapter 10 Resource
Masters.
Simplify each expression. (Lesson 9-1)
39a3b4 3b k  3 10kl 5y  15z y  3z 5
59.   60.    2 61.    
13a4b3 a 5kl k3 42x2 14x 3x
1
62. PHYSICS If a stone is dropped from a cliff, the equation t
 d  represents
4
the time t in seconds that it takes for the stone to reach the ground. If d
represents the distance in feet that the stone falls, find how long it would take
for a stone to fall from a 150-foot cliff. (Lesson 5-6) 3.06 s

Getting Ready for PREREQUISITE SKILL Solve each equation or inequality. Check your solutions.
the Next Lesson (To review solving logarithmic equations and inequalities, see Lesson 10-2.)
63. log3 x
log3 (2x  1) 1 64. log10 2x
log10 32 5
5 3
65. log2 3x log2 5 x   66. log5 (4x  3)  log5 11   x  2
3 4
Answers
546 Chapter 10 Exponential and Logarithmic Relations
47. Let b x  m and by  n. Then
logb m  x and logb n  y.
x m
by  
50. Let bx  m, then logb m  x.
b n
(bx)p  mp
 m
b x y   Quotient Property bxp  mp Product of Powers
n
m logb b  logb m
xp p
logb bx  y  logb  Property of Equality for Logarithmic Equations Property of Equality for Logarithmic Equations
n
m xp  logb mp Inverse Property of Exponents and Logarithms
x  y  logb  Inverse Property of Exponents and Logarithms
n p logb m  logb mp Replace x with logb m.
m
logb m  logb n  logb  Replace x with logb m and y with logb n.
n

546 Chapter 10 Exponential and Logarithmic Relations

0520-573D Alg 2 Ch10-828000 11/22/02 10:56 PM Page 547

Common Logarithms Lesson

Notes

• Solve exponential equations and inequalities using common logarithms.

• Evaluate logarithmic expressions using the Change of Base Formula.
Acidity of Common
1 Focus
Vocabulary is a logarithmic scale used Substances
• common logarithm to measure acidity? Substance pH Level 5-Minute Check
• Change of Base Formula Battery acid 1.0 Transparency 10-4 Use as
The pH level of a substance measures its acidity. A Sauerkraut 3.5
low pH indicates an acid solution while a high pH Tomatoes 4.2 a quiz or review of Lesson 10-3.
Black Coffee 5.0
indicates a basic solution. The pH levels of some
Milk
common substances are shown. Distilled Water
6.4
7.0
Mathematical Background notes
The pH level of a substance is given by Eggs 7.8 are available for this lesson on
Milk of 10.0
pH ⫽ ⫺log10 [H⫹], where H⫹ is the substance’s magnesia p. 520D.
hydrogen ion concentration in moles per liter.
Another way of writing this formula is pH ⫽ ⫺log [H⫹]. is a logarithmic scale
used to measure acidity?
COMMON LOGARITHMS You have seen that the base 10 logarithm function, Ask students:
y ⫽ log10 x, is used in many applications. Base 10 logarithms are called common • What does pH level measure?
logarithms . Common logarithms are usually written without the subscript 10.
the acidity of a substance
log10 x ⫽ log x, x ⬎ 0
• Where have you heard of pH
Study Tip Most calculators have a LOG key for evaluating common logarithms. levels? Sample answer: in soap
Technology and shampoo commercials
Example 1 Find Common Logarithms
Nongraphing scientific • Distilled water has a neutral
calculators often require Use a calculator to evaluate each expression to four decimal places.
entering the number
pH. That is, it is neither acidic
a. log 3 KEYSTROKES: LOG 3 ENTER .4771212547 about 0.4771
followed by the function, nor basic. Use the chart to
for example, 3 LOG . b. log 0.2 KEYSTROKES: LOG 0.2 ENTER –.6989700043 about ⫺0.6990 determine the pH level for a
neutral substance. 7.0
Sometimes an application of logarithms requires that you use the inverse of
logarithms, or exponentiation.
10log x ⫽ x

Example 2 Solve Logarithmic Equations Using Exponentiation

2 Teach
EARTHQUAKES The amount of energy E, in ergs, that an earthquake releases
is related to its Richter scale magnitude M by the equation log E ⫽ 11.8 ⫹ 1.5M.
COMMON LOGARITHMS
The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much
energy was released? Teaching Tip Stress that when the
log E ⫽ 11.8 ⫹ 1.5M Write the formula. base of a logarithm is not shown,
the base is assumed to be 10.
log E ⫽ 11.8 ⫹ 1.5(8.5) Replace M with 8.5.
log E ⫽ 24.55 Simplify.

10log E ⫽ 1024.55 Write each side using exponents and base 10.

E ⫽ 1024.55 Inverse Property of Exponents and Logarithms

E ⬇ 3.55 ⫻ 1024 Use a calculator.

The amount of energy released by this earthquake was about 3.55 ⫻ 1024 ergs.

Lesson 10-4 Common Logarithms 547

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters 5-Minute Check Transparency 10-4
• Study Guide and Intervention, pp. 591–592 Real-World Transparency 10
• Skills Practice, p. 593 Answer Key Transparencies
• Practice, p. 594
• Reading to Learn Mathematics, p. 595 Technology
• Enrichment, p. 596 Interactive Chalkboard

Lesson x-x Lesson Title 547

In-Class Examples Power Example 3 Solve Exponential Equations Using Logarithms
Point®
Solve 3x  11.
1 Use a calculator to evaluate Study Tip 3x
11 Original equation
each expression to four Using Logarithms log 3x
log 11 Property of Equality for Logarithmic Functions
decimal places. When you use the
x log 3
log 11 Power Property of Logarithms
Property for Logarithmic
a. log 6 about 0.7782 Functions as in the second log 11
step of Example 3, this is x
 Divide each side by log 3.
log 3
b. log 0.35 about 0.4559 sometimes referred to as
1.0414
taking the logarithm of x   Use a calculator.
2 EARTHQUAKE Refer to each side. 0.4771

Example 2. The San Fernando x  2.1828 The solution is approximately 2.1828.

Valley earthquake of 1994
CHECK You can check this answer using a calculator or by using estimation.
measured 6.6 on the Richter Since 32
9 and 33
27, the value of x is between 2 and 3. In addition,
scale. How much energy did the value of x should be closer to 2 than 3, since 11 is closer to 9 than 27.
this earthquake release? Thus, 2.1828 is a reasonable solution. ⻫
about 5.01  1021 ergs
Teaching Tip After discussing Example 4 Solve Exponential Inequalities Using Logarithms
In-Class Example 2, have stu- Solve 53y  8y  1.
dents compare the Richter scale 53y  8y  1 Original inequality
magnitudes of the Chilean and
log 53y  log 8y  1 Property of Inequality for Logarithmic Functions
San Fernando Valley earth-
quakes. 8.5  6.6  1.29; the 3y log 5  (y  1) log 8 Power Property of Logarithms
Chilean magnitude was about 3y log 5  y log 8  log 8 Distributive Property
29% greater. Then have them 3y log 5  y log 8  log 8 Subtract y log 8 from each side.
compare the energy released by y(3 log 5  log 8)  log 8 Distributive Property
the Chilean earthquake to the log 8
energy released by the San y   Divide each side by 3 log 5  log 8.
3 log 5  log 8
Fernando Valley earthquake. (0.9031)
3.55  1024  5.01  1021  y   Use a calculator.
3(0.6990)  0.9031
708.58; the Chilean earthquake
y  0.7564 The solution set is {yy  0.7564}.
released more than 6 times as
much energy. Point out that
CHECK Test y
1.
these results demonstrate the
nonlinear nature of the equa- 53y  8y  1 Original inequality
tion that models the amount of 53(1)  8(1)  1 Replace y with 1.
energy released.
53  82 Simplify.

3 Solve 5x
62. about 2.5643 1 1
   ⻫ Negative Exponent Property
125 64

4 Solve 27x 35x  3.

{x | x  5.1415} CHANGE OF BASE FORMULA The Change of Base Formula allows you to
write equivalent logarithmic expressions that have different bases.

Change of Base Formula

• Symbols For all positive numbers, a, b and n, where a  1 and b  1,
b log n ← log base b of original number
loga n
 .
logb a ← log base b of old base

10 log 12
• Example log5 12

log

10 5

548 Chapter 10 Exponential and Logarithmic Relations

Unlocking Misconceptions
Change of Base As you discuss the Change of Base Formula, point
out that the base b that students are changing to does not have to be
10. Any base could be used; however, b is most commonly 10 because
this allows for the logarithms to be evaluated with a calculator.

548 Chapter 10 Exponential and Logarithmic Relations

 To prove this formula, let loga n
x. CHANGE OF BASE
ax
n Definition of logarithm FORMULA
logb ax
logb n Property of Equality for Logarithms
x logb a
logb n Power Property of Logarithms
In-Class Example Power
Point®
log n
x

b
Divide each side by logb a. 5 Express log3 18 in terms of
logb a
logb n common logarithms. Then
loga n
 Replace x with loga n.
logb a approximate its value to four
decimal places. log3 18 
This formula makes it possible to evaluate a logarithmic expression of any base by log10 18
translating the expression into one that involves common logarithms.  ; log3 18  2.6309
log10 3
Example 5 Change of Base Formula
Express log4 25 in terms of common logarithms. Then approximate its value to
four decimal places.

log4 25
1
0 log 25
Change of Base Formula
log10 4
3 Practice/Apply
 2.3219 Use a calculator.

The value of log4 25 is approximately 2.3219.

Study Notebook
Have students—
• add the definitions/examples of
Concept Check 1. Name the base used by the calculator LOG key. What are these logarithms the vocabulary terms to their
called? 10; common logarithms Vocabulary Builder worksheets for
2. Sample answer: 2. OPEN ENDED Give an example of an exponential equation requiring the use Chapter 10.
5x  2; x  0.4307 of logarithms to solve. Then solve your equation.
3. Explain why you must use the Change of Base Formula to find the value of
• include an example of an expo-
log2 7 on a calculator. A calculator is not programmed to find base 2 logarithms. nential inequality that they solved,
and an example showing how to
Guided Practice Use a calculator to evaluate each expression to four decimal places.
use the Change of Base Formula.
GUIDED PRACTICE KEY 4. log 4 0.6021 5. log 23 1.3617 6. log 0.5 0.3010
• include any other item(s) that they
Exercises Examples
Solve each equation or inequality. Round to four decimal places. 12. {pp
4.8188} find helpful in mastering the skills
4–6 1
7–12 3, 4 7. 9x
45 1.7325 8. 45n 30 {nn  0.4907} 9. 3.1a  3
9.42 4.9824 in this lesson.

25.4 1.1615 11. 7t  2
5t 11.5665 12. 4p  1  3p
2
13–15 5 10. 11x
16 2
Express each logarithm in terms of common logarithms. Then approximate its
value to four decimal places.
log 5 log 42 log 9
13. log7 5 ; 0.8271 14. log3 42 ; 3.4022 15. log2 9 ; 3.1699
log 7 lo g 3 log 2

Application 16. DIET Sandra’s doctor has told her to avoid foods with a pH that is less than
About the Exercises…
4.5. What is the hydrogen ion concentration of foods Sandra is allowed to eat? Organization by Objective
Use the information at the beginning of the lesson. at least 0.00003 mole per • Common Logarithms:
liter 17–44, 51–55
★ indicates increased difficulty
• Change of Base Formula:
Practice and Apply 45–50
Use a calculator to evaluate each expression to four decimal places. Odd/Even Assignments
17. log 5 0.6990 18. log 12 1.0792 19. log 7.2 0.8573 Exercises 17–52 are structured
20. log 2.3 0.3617 21. log 0.8 0.0969 22. log 0.03 1.5229 so that students practice the
www.algebra2.com/extra_examples Lesson 10-4 Common Logarithms 549 same concepts whether they
are assigned odd or even
problems.

Differentiated Instruction Assignment Guide

Naturalist Have interested students research earthquakes and their Basic: 17–41 odd, 45, 47, 51,
Richter scale measurements. Have them calculate the amount of energy 56, 59–77
released by three earthquakes they found to be of interest. Have Average: 17–51 odd, 56–77
students compare the energy released to the amount of destruction
caused and share their findings with the class. Advanced: 18–52 even, 53–74
(optional: 72–77)

Lesson 10-4 Common Logarithms 549

 Study
10-4
NAME ______________________________________________ DATE

Guide
Study andIntervention
Guide and Intervention,
____________ PERIOD _____
Homework Help ACIDITY For Exercises 23–26, use the information at the beginning of the lesson
p. 591 (shown) to find the pH of each substance given its concentration of hydrogen ions.
Logarithms and p. 592
For See
Common Exercises Examples
Common Logarithms Base 10 logarithms are called common logarithms. The 23. ammonia: [H]
1 1011 mole per liter 11
expression log10 x is usually written without the subscript as log x. Use the LOG key on 17–22 1
24. vinegar: [H]
6.3 103 mole per liter 2.2
your calculator to evaluate common logarithms.
The relation between exponents and logarithms gives the following identity. 23–44, 3, 4
53–57
25. lemon juice: [H]
7.9 103 mole per liter 2.1
Inverse Property of Logarithms and Exponents 10log x
x

45–50 5
26. orange juice: [H]
3.16 104 mole per liter 3.5
Example 1 Evaluate log 50 to four decimal places.
Use the LOG key on your calculator. To four decimal places, log 50
1.6990. 51–55 2
Example 2 Solve 32x  1  12.
32x  1
12
log 32x  1
log 12
Original equation
Property of Equality for Logarithms
Extra Practice Solve each equation or inequality. Round to four decimal places.
(2x  1) log 3
log 12 See page 850.
27. 6x  42 {xx  2.0860} 28. 5x
52 2.4550
Power Property of Logarithms
log 12
2x  1
 Divide each side by log 3.
log 3

29. 82a  124 {aa  1.1590} 30. 43p

10 0.5537
log 12
2x
  1 Subtract 1 from each side.
log 3

x
 1
1 log 12
 Multiply each side by  .
1

31. 3n  2
14.5 0.4341 32. 9z  4
6.28 4.8362
2 log 3 2

x  0.6309

Exercises 33.
42.5 4.7820 8.2n  3
34. 2.1t  5
9.32 8.0086
log 13
Use a calculator to evaluate each expression to four decimal places.
45.   3.7004 35. 20
70 1.1909
x 2
36. 2x  3
15 2.6281
2
1. log 18
1.2553
2. log 39
1.5911
3. log 120
2.0792
log 2
log 20 37. 8 52
2n 4n  3 {nn  1.0178} 38. 2  3
33x 1.0890
2x
4. log 5.8 5. log 42.3 6. log 0.003
2.5229 46.   1.8614
39. 16d  4
33  d 3.7162 40. 7p  2  135  p {pp
1.9803}
0.7634 1.6263
Lesson 10-4 log 5
log 3 5y  2
22y  1 0.5873 42. 82x  5
5x  1 4.7095
Solve each equation or inequality. Round to four decimal places.

7. 43x
12 0.5975 8. 6x  2
18 0.3869 47.   0.5646 41. 5
log 7
9. 54x  2
120 1.2437 10. 73x  1  21 {x | x  0.8549}
log 8 ★ 43. 2

n 3 n  2 7.6377 ★ 44. 4x
 5x  2 2.7674
11. 2.4x  4
30 0.1150 12. 6.52x  200 {x | x  1.4153} 48.   1.8928
log 3
13. 3.64x  1
85.4 1.1180 14. 2x  5
3x  2 13.9666
2 log 1.6 Express each logarithm in terms of common logarithms. Then approximate its
15. 93x
45x  2 8.1595 16. 6x  5
27x  3 3.6069 49.   0.6781 value to four decimal places.
log 4
Gl

Skills
NAME
/M G

Practice,
______________________________________________
Hill

p. 593 and
591 DATE ____________
GlPERIOD
Al _____
b 2
0.5 log 5 45. log2 13 46. log5 20 47. log7 3
10-4 Practice (Average) 50.   0.4491
Practice, p. 594 (shown)
Common Logarithms log 6 48. log3 8 ★ 49. log4 (1.6)2
★ 50. log6 5
Use a calculator to evaluate each expression to four decimal places.

1. log 101 2.0043 2. log 2.2 0.3424 3. log 0.05 1.3010

For Exercises 51 and 52, use the information presented at the beginning of the
Use the formula pH  log[H] to find the pH of each substance given its
concentration of hydrogen ions. lesson.
4. milk: [H]
2.51 107 mole per liter 6.6

5. acid rain: [H]

2.51 106 mole per liter 5.6
51. POLLUTION The acidity of water determines the toxic effects of runoff into
6. black coffee: [H]
1.0 105 mole per liter 5.0
streams from industrial or agricultural areas. A pH range of 6.0 to 9.0 appears to
7. milk of magnesia: [H]
3.16 1011 mole per liter 10.5 provide protection for freshwater fish. What is this range in terms of the water’s
Solve each equation or inequality. Round to four decimal places. hydrogen ion concentration? between 0.000000001 and 0.000001 mole per liter
8. 2x  25 {x | x  4.6439} 9. 5a
120 2.9746 10. 6z
45.6 2.1319

11. 9m  100 {m | m  2.0959} 12. 3.5x

47.9 3.0885 13. 8.2 y
64.5 1.9802 52. BUILDING DESIGN The 1971 Sylmar earthquake in Los Angeles had a Richter
14. 2b  1  7.31 {b | b
1.8699} 15. 42x
27 1.1887 16. 2a  4
82.1 10.3593 scale magnitude of 6.3. Suppose an architect has designed a building strong
17. 9z  2 38 {z | z  3.6555} 18. 5w  3
17 1.2396 x2
50 1.0725
19. 30
enough to withstand an earthquake 50 times as intense as the Sylmar quake.
3
72 2.3785 21. 42x
9x  1 3.8188 22. 2n  1
52n  1 0.9117
2
20. 5x
Find the magnitude of the strongest quake this building is designed to
Express each logarithm in terms of common logarithms. Then approximate its
value to four decimal places.
Pollution withstand. 8
10 12
log 10 log 32 10 log 9
23. log5 12  ; 1.5440 24. log8 32  ; 1.6667 25. log11 9  ; 0.9163
log10 5 log10 8 log10 11
log10 18 log10 6 log10 8 As little as 0.9 milligram
26. log2 18 
log10 2
; 4.1699 27. log9 6 
log10 9
; 0.8155 28. log7 8
 
2 log10 7
; 0.5343
per liter of iron at a pH of ASTRONOMY For Exercises 53–55, use the following information.
29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H]
5.5 can cause fish to die. Some stars appear bright only because they are very close to us. Absolute
in the soil is not less than 1.58 108 mole per liter. What is the pH of the soil in which
these irises will flourish? 7.8 or less
Source: Kentucky Water Watch magnitude M is a measure of how bright a star would appear if it were 10 parsecs,
30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater is
the hydrogen ion concentration of vinegar than of milk? about 5000
about 32 light years, away from Earth. A lower magnitude indicates a brighter star.
31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubles Absolute magnitude is given by M
m  5  5 log d, where d is the star’s distance
each hour. The number of bacteria N present after t hours is N
1000(2) t. How long will
it take the culture to increase to 50,000 bacteria? about 5.6 h from Earth measured in parsecs and m is its apparent magnitude.
32. SOUND An equation for loudness L in decibels is given by L
10 log R, where R is the
sound’s relative intensity. An air-raid siren can reach 150 decibels and jet engine noise
can reach 120 decibels. How many times greater is the relative intensity of the air-raid
53. Sirius 53. Sirius and Vega are two of the brightest stars in Earth’s sky. The apparent
siren than that of the jet engine noise? 1000 magnitude of Sirius is 1.44 and of Vega is 0.03. Which star appears brighter?
Gl NAME
/M G ______________________________________________
Hill 594 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
10-4 Readingto
to Learn
Learn Mathematics 54. Sirius is 2.64 parsecs from Earth while Vega is 7.76 parsecs from Earth. Find the
Mathematics, p. 595 ELL absolute magnitude of each star. Sirius: 1.45, Vega: 0.58
Common Logarithms
Pre-Activity Why is a logarithmic scale used to measure acidity?
Read the introduction to Lesson 10-4 at the top of page 547 in your textbook.
55. Which star is actually brighter? That is, which has a lower absolute magnitude?
Which substance is more acidic, milk or tomatoes?
tomatoes
Vega
56. CRITICAL THINKING
Reading the Lesson
1
1. Rhonda used the following keystrokes to enter an expression on her graphing calculator: a. Without using a calculator, find the value of log2 8 and log8 2. 3; 
LOG 17 ) ENTER 3 3 2
The calculator returned the result 1.230448921. b. Without using a calculator, find the value of log9 27 and log27 9. ; 
Which of the following conclusions are correct? a, c, and d 2 3
a. The base 10 logarithm of 17 is about 1.2304. c. Make and prove a conjecture as to the relationship between loga b and logb a.
b. The base 17 logarithm of 10 is about 1.2304.
550 Chapter 10 Exponential and Logarithmic Relations
See margin.
c. The common logarithm of 17 is about 1.230449.
d. 101.230448921 is very close to 17.

e. The common logarithm of 17 is exactly 1.230448921.

2. Match each expression from the first column with an expression from the second column
NAME ______________________________________________ DATE ____________ PERIOD _____
that has the same value.

a. log2 2 iv i. log4 1
Enrichment,
10-4 Enrichment p. 596
b. log 12 iii ii. log2 8

c. log3 1 i iii. log10 12

The Slide Rule
1
d. log5  v iv. log5 5
5 Before the invention of electronic calculators, computations were often
e. log 1000 ii v. log 0.1 performed on a slide rule. A slide rule is based on the idea of logarithms. It has
two movable rods labeled with C and D scales. Each of the scales is logarithmic.
3. Calculators do not have keys for finding base 8 logarithms directly. However, you can use
a calculator to find log8 20 if you apply the change of base formula. C 1 2 3 4 5 6 7 8 9
Which of the following expressions are equal to log8 20? B and C
D 1 2 3 4 5 6 7 8 9
log10 20 log 20 log 8
A. log20 8 B.  C.  D. 
log10 8 log 8 log 20
To multiply 2 3 on a slide rule, move the C rod to the right as shown
Helping You Remember below. You can find 2 3 by adding log 2 to log 3, and the slide rule adds the
lengths for you. The distance you get is 0.778, or the logarithm of 6.
4. Sometimes it is easier to remember a formula if you can state it in words. State the
change of base formula in words. Sample answer: To change the logarithm of a log 2 log 3
number from one base to another, divide the log of the original number
in the old base by the log of the new base in the old base. 1 2 3 4 5 6 7 8 9
C
D
1 2 3 4 5 6 7 8 9

550 Chapter 10 Exponential and Logarithmic Relations

 MONEY For Exercises 57 and 58, use the following information.
If you deposit P dollars into a bank account paying an annual interest rate r
(expressed as a decimal), with n interest payments each year, the amount A you 4 Assess
r nt

would have after t years is A
P 1   . Marta places $100 in a savings account
n
earning 6% annual interest, compounded quarterly.
Open-Ended Assessment
57. If Marta adds no more money to the account, how long will it take the money in
Writing Ask students to explain
the account to reach $125? about 3.75 yr or 3 yr 9 mo in writing what it means to use
58. How long will it take for Marta’s money to double? about 11.64 yr or 11 yr 8 mo the Change of Base Formula.
They should include comments
59. WRITING IN MATH Answer the question that was posed at the beginning of about why this formula is useful.
the lesson. See margin.
Why is a logarithmic scale used to measure acidity?
Include the following in your answer:
Getting Ready for
• the hydrogen ion concentration of three substances listed in the table, and Lesson 10-5
• an explanation as to why it is important to be able to distinguish between a PREREQUISITE SKILL In Lesson
hydrogen ion concentration of 0.00001 mole per liter and 0.0001 mole per liter. 10-5, students will solve expo-
nential equations and inequalities
Standardized 60. QUANTITATIVE COMPARISION Compare the quantity in Column A and the using natural logarithms and the
Test Practice quantity in Column B. Then determine whether: A
skills they learned solving com-
A the quantity in Column A is greater, mon logarithmic equations and
B the quantity in Column B is greater, inequalities. Students should be
C the two quantities are equal, or confident when converting be-
D the relationship cannot be determined from the information given. tween exponential and logarith-
Column A Column B mic equations before proceeding.
Use Exercises 72–77 to determine
log 103 log 102
your students’ familiarity with
61. If 24
3x, then what is the value of x? C converting between exponential
A 0.63 B 2.34 C 2.52 D 4 and logarithmic equations.

Maintain Your Skills Answers

Mixed Review 1
Use log7 2  0.3562 and log7 3  0.5646 to approximate the value of each 56c. conjecture: loga b   ;
expression. (Lesson 10-3) logb a
proof:
62. log7 16 1.4248 63. log7 27 1.6938 64. log7 36 1.8416
1
loga b   Original statement
Solve each equation or inequality. Check your solutions. (Lesson 10-2) logb a
66.
z0  z

1
65. log4 r
3 64 66. log8 z  2 67. log3 (4x  5)
5 62 logb b 1
64   Change of Base
logb a logb a Formula
68. Use synthetic substitution to find f(2) for f(x)
x3  6x  2. (Lesson 7-4) 22
1 1
   ✓ Inverse Property of
Factor completely. If the polynomial is not factorable, write prime. (Lesson 5-4) logb a logb a Exponents and
69. 3d2  2d  8 70. 42pq  35p  18q  15 71. 13xyz  3x2z  4k Logarithms
(d  2)(3d  4) (7p  3)(6q  5) prime 59. Comparisons between substances
Getting Ready for PREREQUISITE SKILLS Write an equivalent exponential equation. of different acidities are more
the Next Lesson (For review of logarithmic equations, see Lesson 10-2.)
easily distinguished on a
72. log2 3
x 2x  3 73. log3 x
2 32  x 74. log5 125
3 53  125 logarithmic scale. Answers should
include the following.
Write an equivalent logarithmic equation.
(For review of logarithmic equations, see Lesson 10-2.) • Sample answer:
75. 5
45 log5 45  x
x 76. 7
x log7 x  3
3 77. b
x logb x  y
y
Tomatoes: 6.3  105 mole
www.algebra2.com/self_check_quiz Lesson 10-4 Common Logarithms 551 per liter
Milk: 3.98  107 mole per liter
Eggs: 1.58  108 mole per liter
• Those measurements correspond to pH measurements
of 5 and 4, indicating a weak acid and a stronger acid.
On the logarithmic scale we can see the difference in
these acids, whereas on a normal scale, these
hydrogen ion concentrations would appear nearly the
same. For someone who has to watch the acidity of the
foods they eat, this could be the difference between
an enjoyable meal and heartburn.

Lesson 10-4 Common Logarithms 551

Graphing
Calculator A Follow-Up of Lesson 10-4
Investigation
A Follow-Up of Lesson 10-4

Getting Started Solving Exponential and Logarithmic

Equations and Inequalities
Using Parentheses In Step 1 of
Example 1, remind students that You can use a TI-83 Plus graphing calculator to solve exponential and
logarithmic equations and inequalities. This can be done by graphing
they must also use parentheses each side of the equation separately and using the intersect feature on
1
around the fraction . the calculator.
2

Example 1
Teach Solve 23x  9  
1 x3
2
by graphing.
• Before discussing Example 1,
use a simple equation such as Graph each side of the equation. Use the intersect feature.
2x
6 to show students how • Graph each side of the equation as a separate • You can use the intersect feature on the CALC
x3 menu to approximate the ordered pair of the
the equation can be solved by 1
function. Enter 23x  9 as Y1. Enter  2 point at which the curves cross.
graphing. Graph the equations as Y2. Be sure to include the added parentheses KEYSTROKES: See page 115 to review how to use
y
2x and y
6 and then the intersect feature.
around each exponent. Then graph the two
identify the point of intersection equations.
of the graphs. KEYSTROKES: See pages 87 and 88 to review
• Ask students why it is necessary graphing equations.
in Step 1 to enter the equations
using parentheses around the
exponents.
• Have students substitute the [2, 8] scl: 1 by [2, 8] scl: 1

solution to Example 1 into the The calculator screen shows that the
original equation to verify that x-coordinate of the point at which the curves
it is correct. [2, 8] scl: 1 by [2, 8] scl: 1 cross is 3. Therefore, the solution of the
equation is 3.
• In Example 2, make sure
students understand why the
The TI-83 Plus has y
log10 x as a built-in function. Enter
equations must be rewritten
using the Change of Base LOG X,T,,n GRAPH to view this graph. To graph y  log10x
logarithmic functions with bases other than 10, you must
Formula. use the Change of Base Formula,
• Students can find the solution loga n

b
.
log n
set for Example 2 without using logb a
the shading options. Simply log x
For example, log3 x
1
0
, so to graph y
log3 x you [2, 8] scl: 1 by [5, 5] scl: 1
log10 3
have them use the intersect
feature, noting that the graph must enter LOG X,T,,n )  LOG 3 ) as Y1.
of Y1 intersects or is above the
graph of Y2 at and to the right www.algebra2.com/other_calculator_keystrokes
of x
0.5.
552 Chapter 10 Exponential and Logarithmic Relations

552 Chapter 10 Exponential and Logarithmic Relations

 Example 2
Solve log2 2x  log12 2x by graphing. Assess
Rewrite the problem as a system of Enter the first inequality. In Exercise 9, check that students
common logarithmic inequalities. log 2x
• Enter y   as Y1. Since the inequality record the inequalities in the solu-
• The first inequality is log2 2x  y or log 2
y  log2 2x. The second inequality is includes less than, shade below the curve. tion set correctly. In particular,
y  log12 2x. students must include the fact
KEYSTROKES: LOG 2 X,T,,n ) 
• Use the Change of Base Formula to create
that x must be greater than 0.
LOG 2 )
equations that can be entered into the
calculator. Use the arrow and ENTER keys to choose the
log 2x log 2x shade below icon, .
log2 2x
 log12 2x

log 2 1
log 
2
log 2x
Thus, the two inequalities are y   and
log 2
log 2x
y  .
1
log 
2

Enter the second inequality. Graph the inequalities.

log 2x
• Enter y   as Y2. Since the inequality KEYSTROKES: GRAPH
1
log 
2
includes greater than, shade above the curve.

KEYSTROKES: LOG 2 X,T,,n ) 

LOG 1  2 ) GRAPH

Use the arrow and ENTER keys to choose the

shade above icon, .
[2, 8] scl: 1 by [5, 5] scl: 1

The x values of the points in the region where

the shadings overlap is the solution set of the
original inequality. Using the calculator’s
intersect feature, you can conclude that the
solution set is {xx  0.5}.

Exercises Solve each equation or inequality by graphing. 7. x  6

1. 3.5x  2
1.75x  3 1.2 2. 3x  4
0.52x  3 2.6 3. 62  x  4
0.25x  2.5 1.8
x
4. 3x  4
5 2 2 5. log2 3x
log3 (2x  2) 0.7 6. 2x  2  0.5x  3 x  2.5
7. log3 (3x  5)  log3 (x  7) 8. 5x  3  2x  4 x
2.24 9. log2 2x  log4 (x  3) 0  x
1

Graphing Calculator Investigation Solving Exponential and Logarithmic Equations and Inequalities 553

Graphing Calculator Investigation Solving Exponential and Logarithmic Equations and Inequalities 553
 Lesson Base e and
Notes Natural Logarithms
• Evaluate expressions involving the natural base and natural logarithms.

1 Focus • Solve exponential equations and inequalities using natural logarithms.

Vocabulary is the natural base e used in banking?

5-Minute Check • natural base, e Suppose a bank compounds interest
Transparency 10-5 Use as • natural base exponential on accounts continuously, that is, with
Continuously Compounded Interest
a quiz or review of Lesson 10-4. function no waiting time between interest A
nt
• natural logarithm payments. In order to develop an
1+ n)
– r

Mathematical Background notes • natural logarithmic equation to determine continuously A= P ( ) 1(1

2
.
14..
1+ 1) )
n 1–
function compounded interest, examine what 2.44
are available for this lesson on 1 ) 1( 4(1
30..
.
happens to the value A of an account
(1+ 4) 12(1)
rly 1–
(yea 2.61
p. 520D. for increasingly larger numbers of 4 ly) 1
45..
.
+ )
1
rter 2.71
(qua 2 1 (1 12 365(1)
compounding periods n. Use a 1 ly) 1 )
81..
.
is the natural base e principal P of $1, an interest rate r of (mo 5
nth + 5
1 (1 36 876
0(1) 2.71
36 ) 1 )
y
used in banking? 100% or 1, and time t of 1 year. (dail
0 1 (1 87
+ 60
876 ly)
r
Ask students: (hou

• What is the formula

calculating? the amount of BASE e AND NATURAL LOGARITHMS In the table above, as n increases,
money in the account 1 n(1) 1 n
the expression 11    
or 1   approaches the irrational number
n n
• Why does the interest increase
2.71828... . This number is referred to as the natural base, e .
as the time between compound-
An exponential function with base e is called a y
ing periods decreases? Sample natural base exponential function . The graph of
answer: Interest is earned not just on y
ex is shown at the right. Natural base exponential
y  ex
the initial $1 but also on the total functions are used extensively in science to model
e1  e
interest that has accrued. As the
Study Tip quantities that grow and decay continuously. (1, e )

Most calculators have an ex function for evaluating e0  1

compounding occurs more often, the Simplifying (0, 1)
Expressions with e natural base expressions.
amount of money earning interest You can simplify
O x
grows faster. expressions involving e in
the same manner in which
you simplify expressions
involving . Example 1 Evaluate Natural Base Expressions
Examples:
Use a calculator to evaluate each expression to four decimal places.
• 2  3
5
• e2  e3
e5 a. e2 KEYSTROKES: 2nd [e x ] 2 ENTER 7.389056099 about 7.3891

b. e1.3 KEYSTROKES: 2nd [e x ] 1.3 ENTER .272531793 about 0.2725

The logarithm with base e is called the y

y  ex
natural logarithm , sometimes denoted by
loge x, but more often abbreviated ln x. The
natural logarithmic function , y
ln x, is the yx
(1, e )
inverse of the natural base exponential function,
y
ex. The graph of these two functions (0, 1)
shows that ln 1
0 and ln e
1. (e , 1)
O (1, 0) x
y  ln x

554 Chapter 10 Exponential and Logarithmic Relations

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters Science and Mathematics Lab Manual, 5-Minute Check Transparency 10-5
• Study Guide and Intervention, pp. 597–598 pp. 127–132 Answer Key Transparencies
• Skills Practice, p. 599
• Practice, p. 600 Technology
• Reading to Learn Mathematics, p. 601 Alge2PASS: Tutorial Plus, Lesson 19
• Enrichment, p. 602 Interactive Chalkboard
• Assessment, p. 624
 Most calculators have an LN key for evaluating natural logarithms.

Example 2 Evaluate Natural Logarithmic Expressions

2 Teach
Use a calculator to evaluate each expression to four decimal places. BASE e AND NATURAL
a. ln 4 KEYSTROKES: LN 4 ENTER 1.386294361 about 1.3863 LOGARITHMS
b. ln 0.05 KEYSTROKES: LN 0.05 ENTER –2.995732274 about 2.9957
Teaching Tip Stress that e is a
constant like , and not a variable
You can write an equivalent base e exponential equation for a natural logarithmic
like x or y.
equation and vice versa by using the fact that ln x
loge x.

Example 3 Write Equivalent Expressions In-Class Examples Power

Point®
Write an equivalent exponential or logarithmic equation.
a. ex  5 b. ln x  0.6931 1 Use a calculator to evaluate
ex
5 → loge 5
x ln x  0.6931 → loge x  0.6931 each expression to four
ln 5
x x e0.6931 decimal places.
a. e 0.5 about 1.6487
Since the natural base function and the natural logarithmic function are inverses,
these two functions can be used to “undo” each other.
b. e8 about 0.0003
eln x
x ln ex
x 2 Use a calculator to evaluate
Example 4 Inverse Property of Base e and Natural Logarithms each expression to four
decimal places.
Evaluate each expression.
a. eln 7 b. ln e4x  3
a. ln 3 about 1.0986
eln 7
7 ln e4x  3
4x  3 1
b. ln  about 1.3863
4

EQUATIONS AND INEQUALITIES WITH e AND ln Equations and 3 Write an equivalent exponen-
inequalities involving base e are easier to solve using natural logarithms than using tial or logarithmic equation.
common logarithms. All of the properties of logarithms that you have learned apply
to natural logarithms as well. a. e x
23 ln 23  x
b. ln x  1.2528 x  e1.2528
Example 5 Solve Base e Equations
Solve 5ex  7  2. 4 Evaluate each expression.
5ex 7
2 Original equation a. eln 21 21
5ex
9 Add 7 to each side. 2
1
b. ln e x x2  1
9
ex
 Divide each side by 5.
5
9
ln ex
ln  Property of Equality for Logarithms
5
9
x
ln  Inverse Property of Exponents and Logarithms
EQUATIONS AND
5
INEQUALITIES WITH e
9
x
ln 
5
Divide each side by 1. AND ln
x  0.5878 Use a calculator.
In-Class Example Power
Point®
The solution is about 0.5878.
5 Solve 3e2x  4
10.
CHECK You can check this value by substituting
0.5878 into the original equation or by x  0.3466
finding the intersection of the graphs of
y
5ex  7 and y
2.

www.algebra2.com/extra_examples Lesson 10-5 Base e and Natural Logarithms 555

Lesson 10-5 Base e and Natural Logarithms 555

 Study Tip When interest is compounded continuously, the amount A in an account after
In-Class Examples Power
Point® t years is found using the formula A
Pert, where P is the amount of principal
Continuously and r is the annual interest rate.
6 SAVINGS Suppose you Compounded
Interest
deposit $700 into an account Although no banks Example 6 Solve Base e Inequalities
paying 6% annual interest, actually pay interest
SAVINGS Suppose you deposit $1000 in an account paying 5% annual interest,
compounded continuously,
compounded continuously. the equation A = Pert is so
compounded continuously.
a. What is the balance after accurate in computing the a. What is the balance after 10 years?
amount of money for
8 years? $1131.25 quarterly compounding, A
Pert Continuous compounding formula

b. How long will it take for the or daily compounding,

1000e(0.05)(10) Replace P with 1000, r with 0.05, and t with 10.
that it is often used for
balance in your account to
1000e0.5 Simplify.
this purpose.
reach at least $2000? at least  1648.72 Use a calculator.

17.5 years The balance after 10 years would be $1648.72.

7 Solve each equation or b. How long will it take for the balance in your account to reach at
least $1500?
inequality.
The balance is at least $1500.
a. ln 3x
0.5 about 0.5496




A  1500 Write an inequality.
b. ln (2x  3)  2.5 1000e(0.05)t  1500 Replace A with 1000e(0.05)t.
1.5  x  7.5912
e(0.05)t  1.5 Divide each side by 1000.
ln e(0.05)t  ln 1.5 Property of Equality for Logarithms
0.05t  ln 1.5 Inverse Property of Exponents and Logarithms
ln 1.5
t   Divide each side by 0.05.
0.05
t  8.11 Use a calculator.
It will take at least 8.11 years for the balance to reach $1500.

Study Tip Example 7 Solve Natural Log Equations and Inequalities

Equations with ln Solve each equation or inequality.
As with other logarithmic
equations, remember to
a. ln 5x  4
check for extraneous ln 5x
4 Original equation
solutions.
eln 5x
e4 Write each side using exponents and base e.
5x
e4 Inverse Property of Exponents and Logarithms
e4
x
 Divide each side by 5.
5
x  10.9196 Use a calculator.

The solution is 10.9196. Check this solution using substitution or graphing.

b. ln (x  1)  2
ln (x  1) 2 Original inequality
eln (x  1) e2 Write each side using exponents and base e.
x  1 e2 Inverse Property of Exponents and Logarithms
x e2  1 Add 1 to each side.
x 1.1353 Use a calculator.

The solution is all numbers greater than about 1.1353. Check this solution using
substitution.

556 Chapter 10 Exponential and Logarithmic Relations

Differentiated Instruction
Kinesthetic Using plastic coins and paper currency, have pairs of
students begin with $10, choose an interest rate, and calculate how
much they will have after 5, 10, 15, and 20 years. After each
calculation, have students model the amount with their money to help
them visualize the growth over time.

556 Chapter 10 Exponential and Logarithmic Relations

 Concept Check 1. Name the base of natural logarithms. the number e
3 Practice/Apply
2. OPEN ENDED Give an example of an exponential equation that requires using
natural logarithms instead of common logarithms to solve. Sample answer:
3. Elsu; Colby tried to 3. FIND THE ERROR Colby and Elsu are solving ln 4x
5. ex  8
write each side as a Study Notebook
power of 10. Since the Colby Elsu Have students—
base of the natural
logarithmic function is ln 4x = 5 ln 4x = 5 • add the definitions/examples of
e, he should have the vocabulary terms to their
written each side as a 10ln 4x = 105 el n 4x = e5
power of e; 10ln 4x  Vocabulary Builder worksheets for
4x = 100,000 4x = e5
4x. 5 Chapter 10.
x = 25,000 x = e
4 • include examples of how to
x ≈ 37.1033 evaluate expressions containing
Who is correct? Explain your reasoning.
natural logarithms.
• include any other item(s) that they
Guided Practice Use a calculator to evaluate each expression to four decimal places.
find helpful in mastering the skills
GUIDED PRACTICE KEY 4. e6 403.4288 5. e3.4 0.0334 6. ln 1.2 0.1823 7. ln 0.1 2.3026
in this lesson.
Exercises Examples Write an equivalent exponential or logarithmic equation.
47 1, 2 8. ex
4 x  ln 4 9. ln 1
0 e0  1
8, 9 3
10, 11 4 Evaluate each expression.
1217 57 10. eln 3 3 11. ln e5x 5x
18, 19 5
Solve each equation or inequality. 15. 0  x  403.4288 FIND THE ERROR
12. ex 30 x  3.4012 13. 2ex  5
1 1.0986 14. 3  e2x
8 0.8047 Make sure stu-
15. ln x  6 16. 2 ln 3x  1
5 2.4630 17. ln x2
9 90.0171 dents can identify what
Application ALTITUDE For Exercises 18 and 19, use the following information. Colby did incorrectly. Point out
The altimeter in an airplane gives the altitude or height h (in feet) of a plane above that raising both terms to base 10
sea level by measuring the outside air pressure P (in kilopascals). The height and air is not incorrect but the step that
h
pressure are related by the model P
101.3 e 26,200 follows incorrectly states that
18. h  18. Find a formula for the height in terms of the outside air pressure. 10ln 4x
4x.
P
26,200 ln 101.3 19. Use the formula you found in Exercise 18 to approximate the height of a plane
above sea level when the outside air pressure is 57 kilopascals. about 15,066 ft

★ indicates increased difficulty About the Exercises…

Practice and Apply Organization by Objective
• Base e and Natural
Homework Help Use a calculator to evaluate each expression to four decimal places. Logarithms: 20–37
For
Exercises
See
Examples
20. e4 54.5982 21. e5 148.4132 22. e1.2 0.3012 23. e0.5 1.6487 • Equations and Inequalities
20–29 1, 2 24. ln 3 1.0986 25. ln 10 2.3026 26. ln 5.42 1.6901 27. ln 0.03 3.5066 with e and ln: 38–61
30–33 3
34–37 4 28. SAVINGS If you deposit $150 in a savings account paying 4% interest Odd/Even Assignments
38–53 5–7 compounded continuously, how much money will you have after 5 years? Exercises 20–53 are structured
54–57 6 Use the formula presented in Example 6. $183.21
58–61 3, 5 I so that students practice the
29. PHYSICS The equation ln 0
0.014d relates the intensity of light at a depth of same concepts whether they
I
Extra Practice d centimeters of water I with the intensity in the atmosphere I0. Find the depth of are assigned odd or even
See page 850. the water where the intensity of light is half the intensity of the light in the
atmosphere. about 49.5 cm
problems.
www.algebra2.com/self_check_quiz Lesson 10-5 Base e and Natural Logarithms 557
Assignment Guide
Basic: 21–51 odd, 54–59, 62–80
Average: 21–53 odd, 54–59,
62–80
Advanced: 20–52 even, 54–74
(optional: 75–80)
All: Practice Quiz 2 (1–5)

Lesson 10-5 Base e and Natural Logarithms 557

 Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Write an equivalent exponential or logarithmic equation.
10-5 Study Guide and
p. 597
Base e (shown) and p. 598
and Natural Logarithms 30. ex
5 31. e2
6x 32. ln e
1 33. ln 5.2
x
Base e and Natural Logarithms The irrational number e  2.71828… often occurs
as the base for exponential and logarithmic functions that describe real-world phenomena.
x  ln 5 2  ln 6x e1  e ex  5.2

As n increases, 1   
1 n
approaches e  2.71828….
Evaluate each expression.
Natural Base e n

34. eln 0.2 0.2 35. eln y y 36. ln e4x 4x 37. ln e45 45
ln x
loge x

The functions y
ex and y
ln x are inverse functions.

Inverse Property of Base e and Natural Logarithms eln x

x ln ex
x

Natural base expressions can be evaluated using the ex and ln keys on your calculator.
Solve each equation or inequality.
Example 1 Evaluate ln 1685.
Use a calculator. 38. 3ex  1
5 0.2877 39. 2ex  1
0 0.6931 40. ex  4.5 x  1.5041
ln 1685  7.4295

Example 2 Write a logarithmic equation equivalent to e 2x  7. 41. ex 1.6 x  0.4700 42. 3e4x  11
2 0.2747 43. 8  3e3x
26 0.5973
44. e5x  25 x  0.6438 45. e2x  7 x  0.9730 46. ln 2x
4 27.2991
e2x
7 → loge 7
2x or 2x
ln 7

Example 3 Evaluate ln e18.

Use the Inverse Property of Base e and Natural Logarithms.
ln e18
18
47. ln 3x
5 49.4711 48. ln (x  1)
1 1.7183 49. ln (x  7)
2 14.3891
Exercises 50. ln x  ln 3x
12 232.9197 51. ln 4x  ln x
9 45.0086
Use a calculator to evaluate each expression to four decimal places. Money ★ 52. ln (x2  12)
ln x  ln 8 2, 6 ★ 53. ln x  ln (x  4)
ln 5 1
1. ln 732 2. ln 84,350 3. ln 0.735 4. ln 100
6.5958 11.3427 0.3079 4.6052 To determine the doubling
5. ln 0.0824 6. ln 2.388 7. ln 128,245 8. ln 0.00614 time on an account paying MONEY For Exercises 54–57, use the formula for continuously compounded
2.4962 5.0929
0.8705 11.7617
an interest rate r that is interest found in Example 6. 55. t   100 In 2
Write an equivalent exponential or logarithmic equation. compounded annually, r
9. e15
x 10. e3x
45 11. ln 20
x 12. ln x
8 investors use the “Rule of 54. If you deposit $100 in an account paying 3.5% interest compounded
ln x  15 3x  ln 45 e x  20 x  e8
72.” Thus, the amount of continuously, how long will it take for your money to double? about 19.8 yr
13. e5x
0.2 14. ln (4x)
9.6 15. e8.2
10x 16. ln 0.0002
x
5x  ln 0.2 4x  e 9.6 ln 10x  8.2 e x  0.0002 time needed for the money 55. Suppose you deposit A dollars in an account paying an interest rate r as a
Lesson 10-5

in an account paying 6%
Evaluate each expression. percent, compounded continuously. Write an equation giving the time t needed
interest compounded
17. ln e3 18. eln 42 19. eln 0.5 20. ln e16.2
72 for your money to double, or the doubling time.
3 42 0.5 16.2
annually to double is  or
Gl NAME
/M G ______________________________________________
Hill 597 DATE ____________
GlPERIOD
Al _____
b 2 6 56. Explain why the equation you found in Exercise 55 might be referred to as the
Skills Practice, p. 599 and 12 years.
10-5 Practice (Average) “Rule of 70.” 100 In 2  70
Practice,
Base e andp. 600Logarithms
Natural (shown) Source: www.datachimp.com
Use a calculator to evaluate each expression to four decimal places.
1. e1.5 4.4817 2. ln 8 2.0794 3. ln 3.2 1.1632 4. e0.6 0.5488
57. MAKE A CONJECTURE State a rule that could be used to approximate the
5. e4.2 66.6863 6. ln 1 0 7. e2.5 0.0821 8. ln 0.037 3.2968 amount of time t needed to triple the amount of money in a savings account
Write an equivalent exponential or logarithmic equation. paying r percent interest compounded continuously. t   11 0
9. ln 50
x
e x  50
10. ln 36
2x
e 2x  36
11. ln 6  1.7918
e1.7918  6
12. ln 9.3  2.2300
e 2.2300  9.3
r
13. ex
8 14. e5
10x 15. ex
4 16. e2
x  1 POPULATION For Exercises 58 and 59, use the following information.
x  ln 8 5  ln 10x x  ln 4 2  ln (x  1)
In 2000, the world’s population was about 6 billion. If the world’s population
Evaluate each expression.
17. eln 12 12 18. eln 3x 3x 19. ln e1 1 20. ln e2y 2y continues to grow at a constant rate, the future population P, in billions, can be
Solve each equation or inequality. predicted by P
6e0.02t, where t is the time in years since 2000. 58. about 7.33 billion
21. ex  9 22. ex
31 23. ex
1.1 24. ex
5.8
{x | x  2.1972} 3.4340 0.0953 1.7579 58. According to this model, what will the world’s population be in 2010?
25. 2ex 3
1 26. 5ex 17 27. 4  ex
19 28. 3ex  10  8
0.6931 {x | x  0.1823} 2.7081 {x | x  0.4055} 59. Some experts have estimated that the world’s food supply can support a
29. e3x
8 30. e4x
5 31. e0.5x
6 32. 2e5x
24
0.6931 0.4024 3.5835 0.4970
population of, at most, 18 billion. According to this model, for how many more
33. e2x  1
55 34. e3x  5
32 35. 9  e2x
10 36. e3x  7  15 years will the world’s population remain at 18 billion or less? about 55 yr
1.9945 1.2036 0 {x | x
0.6931}
37. ln 4x
3
5.0214
38. ln (2x)
7
548.3166
39. ln 2.5x
10
8810.5863
40. ln (x  6)
1
8.7183 Online Research Data Update What is the current world population?
41. ln (x  2)
3 42. ln (x  3)
5 43. ln 3x  ln 2x
9 44. ln 5x  ln x
7 Visit www.algebra2.com/data_update to learn more.
18.0855 145.4132 36.7493 14.8097
INVESTING For Exercises 45 and 46, use the formula for continuously
compounded interest, A  Pert, where P is the principal, r is the annual interest
rate, and t is the time in years. RUMORS For Exercises 60 and 61, use the following information.
45. If Sarita deposits $1000 in an account paying 3.4% annual interest compounded
continuously, what is the balance in the account after 5 years? $1185.30
The number of people H who have heard a rumor can be approximated by
P
46. How long will it take the balance in Sarita’s account to reach $2000? about 20.4 yr
H

0.35t , where P is the total population, S is the number of people
47. RADIOACTIVE DECAY The amount of a radioactive substance y that remains after 1  (P  S)e
t years is given by the equation y
aekt, where a is the initial amount present and k is
the decay constant for the radioactive substance. If a
100, y
50, and k
0.035,
find t. about 19.8 yr
who start the rumor, and t is the time in minutes. Suppose two students start a
Gl NAME
/M G ______________________________________________
Hill 600 DATE ____________
Gl PERIOD
Al _____
b 2
rumor that the principal will let everyone out of school one hour early that day.
Reading
10-5 Readingto
to Learn
Learn Mathematics
Mathematics, p. 601 ELL 60. If there are 1600 students in the school, how many students will have heard the
Base e and Natural Logarithms
Pre-Activity How is the natural base e used in banking?
rumor after 10 minutes? about 32 students
Read the introduction to Lesson 10-5 at the top of page 554 in your textbook.
Suppose that you deposit $675 in a savings account that pays an annual
61. How much time will pass before half of the students have heard the rumor?
interest rate of 5%. In each case listed below, indicate which method of
compounding would result in more money in your account at the end of one
about 21 min
year.
a. annual compounding or monthly compounding monthly
62. CRITICAL THINKING Determine whether the following statement is sometimes,
b. quarterly compounding or daily compounding daily
c. daily compounding or continuous compounding continuous
always, or never true. Explain your reasoning. Always; see pp. 573A-573D.
log x ln x
Reading the Lesson For all positive numbers x and y, 
.
1. Jagdish entered the following keystrokes in his calculator:
log y ln y
LN 5 ) ENTER

The calculator returned the result 1.609437912. Which of the following conclusions are 558 Chapter 10 Exponential and Logarithmic Relations
correct? d and f
a. The common logarithm of 5 is about 1.6094.
b. The natural logarithm of 5 is exactly 1.609437912.
c. The base 5 logarithm of e is about 1.6094.
d. The natural logarithm of 5 is about 1.609438. NAME ______________________________________________ DATE ____________ PERIOD _____

e. 101.609437912 is very close to 5.

f. e1.609437912 is very close to 5. Enrichment,
10-5 Enrichment p. 602
2. Match each expression from the first column with its value in the second column. Some
choices may be used more than once or not at all. Approximations for  and e
a. eln 5 IV I. 1 The following expression can be used to approximate e. If greater and greater
b. ln 1 V II. 10 values of n are used, the value of the expression approximates e more and
more closely.
c. eln e VI III. 1
1  n1
n
d. ln e5 IV IV. 5
e. ln e I V. 0 Another way to approximate e is to use this infinite sum. The greater the

  III
1 value of n, the closer the approximation.
f. ln  VI. e
e 1 1 1 1
e
1  1        …    …
2 23 234 234…n
Helping You Remember In a similar manner,  can be approximated using an infinite product
discovered by the English mathematician John Wallis (1616–1703).
3. A good way to remember something is to explain it to someone else. Suppose that you are
studying with a classmate who is puzzled when asked to evaluate ln e3. How would you  2 2 4 4 6 6 2n 2n

            …     …
explain to him an easy way to figure this out? Sample answer: ln means natural 2 1 3 3 5 5 7 2n  1 2n  1
log. The natural log of e 3 is the power to which you raise e to get e 3. This
is obviously 3. Solve each problem.

1. Use a calculator with an ex key to find e to 7 decimal places. 2.7182818

558 Chapter 10 Exponential and Logarithmic Relations

 63. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See margin.
How is the natural base e used in banking?
4 Assess
Include the following in your answer:
Open-Ended Assessment
• an explanation of how to calculate the value of an account whose interest is
compounded continuously, and Speaking Ask students to ex-
• an explanation of how to use natural logarithms to find when the account will plain how evaluating expressions
have a specified value. involving base e and natural
2 1
logarithms is similar to evalu-
Standardized 64. If ex  1 and ex

2x,
what is the value of x? B ating expressions involving
Test Practice common logarithms and base 10,
A 1.41 B 0.35 C 1.00 D 1.10
and also how they differ.
65. SHORT RESPONSE The population of a certain country can be modeled by the
equation P(t)
40 e0.02t, where P is the population in millions and t is the
number of years since 1900. When will the population be 100 million, 200 Getting Ready for
million, and 400 million? What do you notice about these time periods? 1946,
1981, 2015; It takes between 34 and 35 years for the population to double. Lesson 10-6
PREREQUISITE SKILL Students
Maintain Your Skills will encounter exponential
growth and decay problems in
Mixed Review Express each logarithm in terms of common logarithms. Then approximate its Lesson 10-6. They will be
value to four decimal places. (Lesson 10-4) required to solve exponential
log 68
66.   3.0437 66. log4 68 67. log6 0.047 68. log50 23 equations and inequalities. Use
log 4
log 0.047 Solve each equation. Check your solutions. (Lesson 10-3) Exercises 75–80 to determine
67.   your students’ familiarity with
log 6
69. log3 (a  3)  log3 (a  3)
log3 16 5 70. log11 2  2 log11 x
log11 32 4
1.7065 solving exponential equations
log 23
State whether each equation represents a direct, joint, or inverse variation. Then and inequalities.
68.   0.8015 name the constant of variation. (Lesson 9-4)
log 50 a
71. mn
4 inverse, 4 72. 
c joint, 1 73. y
7x direct, 7
b Assessment Options
74. COMMUNICATION A microphone is placed at the focus of a parabolic reflector Practice Quiz 2 The quiz
to collect sounds for the television broadcast of a football game. The focus of the
provides students with a brief
parabola that is the cross section of the reflector is 5 inches from the vertex. The
latus rectum is 20 inches long. Assuming that the focus is at the origin and the review of the concepts and skills
parabola opens to the right, write the equation of the cross section. (Lesson 8-2) in Lessons 10-3 through 10-5.
1 Lesson numbers are given to the
x  y 2  5
Getting Ready for 20
PREREQUISITE SKILL Solve each equation or inequality. right of the exercises or instruc-
the Next Lesson (To review exponential equations and inequalities, see Lesson 10-1.)
tion lines so students can review
75. 2x
10 3.32 76. 5x
12 1.54 77. 6x
13 1.43 concepts not yet mastered.
78. 2(1  0.1)x
50 79. 10(1  0.25)x
200 80. 400(1  0.2)x
50 9.32
323.49 13.43 Quiz (Lessons 10-4 and 10-5) is
available on p. 624 of the
Chapter 10 Resource Masters.
P ractice Quiz 2 Lessons 10-3 through 10-5
1. Express log4 5 in terms of common logarithms. Then approximate its value to
four decimal places. (Lesson 10-4) log 5
; 1.1610
log 4
2. Write an equivalent exponential equation for ln 3x
2. (Lesson 10-5) e2  3x

Solve each equation or inequality. (Lesson 10-3 through 10-5)

3. log2 (9x  5)
2  log2 (x2  1) 3 4. 2x  3 5 x  5.3219 5. 2ex  1
7 1.3863

Lesson 10-5 Base e and Natural Logarithms 559

Answer
63. The number e is used in the formula for continuously • If you know the annual interest • If you know the value A you wish
compounded interest, A  Pe rt. Although no banks rate r and the principal P, the the account to achieve, the princi-
actually pay interest compounded continually, the value of the account after t years pal P, and the annual interest rate
equation is so accurate in computing the amount of is calculated by multiplying P r, the time t needed to achieve this
money for quarterly compounding, or daily times e raised to the r times t value is found by first taking the
compounding, that it is often used for this purpose. power. Use a calculator to find natural logarithm of A minus the
Answers should include the following. the value of ert. natural logarithm of P. Then,
divide this quantity by r.
Lesson 10-5 Base e and Natural Logarithms 559
 Lesson Exponential Growth and Decay
Notes

• Use logarithms to solve problems involving exponential decay.

1 Focus • Use logarithms to solve problems involving exponential growth.

Vocabulary can you determine the current value of your car?

5-Minute Check • rate of decay Certain assets, like homes, can
Transparency 10-6 Use as • rate of growth appreciate or increase in value over
a quiz or review of Lesson 10-5. time. Others, like cars, depreciate
or decrease in value with time. Years after Value of
Purchase Car ($)
Mathematical Background notes Suppose you buy a car for $22,000
and the value of the car decreases 0 22,000.00
are available for this lesson on by 16% each year. The table 1 18,480.00
p. 520D. shows the value of the car each 2 15,523.20
year for up to 5 years after it was 3 13,039.49
can you determine the purchased. 4 10,953.17
current value of your 5 9200.66
car?
Ask students:
• What kinds of items increase in
value? Sample answer: artwork,
some trading cards, some EXPONENTIAL DECAY The depreciation of the value of a car is an example of
collectibles exponential decay. When a quantity decreases by a fixed percent each year, or other
period of time, the amount y of that quantity after t years is given by y
a(1  r)t,
• During which year does the car where a is the initial amount and r is the percent of decrease expressed as a decimal.
depreciate the most? first year The percent of decrease r is also referred to as the rate of decay .
• How can the amount of depre-
ciation be different each year Example 1 Exponential Decay of the Form y  a(1  r)t
when the percent of decrease is CAFFEINE A cup of coffee contains 130 milligrams of caffeine. If caffeine is
always the same? In the first eliminated from the body at a rate of 11% per hour, how long will it take for
year, the car depreciates 16% of its
Study Tip half of this caffeine to be eliminated from a person’s body?

value at the beginning of that year Rate of Change Explore The problem gives the amount of caffeine consumed and the rate at
Remember to rewrite
(when it was new). This is when its the rate of change as a
which the caffeine is eliminated. It asks you to find the time it will take
value is greatest, so the amount of decimal before using it for half of the caffeine to be eliminated from a person’s body.
depreciation is also the greatest in the formula.
Plan Use the formula y
a(1  r)t. Let t be the number of hours since
during this year. The amount of drinking the coffee. The amount remaining y is half of 130 or 65.
depreciation decreases each year Solve y
a(1  r)t Exponential decay formula
because the value of the car at the
65
130(1  0.11)t Replace y with 65, a with 130, and r with 11% or 0.11.
beginning of each year is less than
it was at the beginning of the 0.5
(0.89)t Divide each side by 130.

previous year. log 0.5

log (0.89)t Property of Equality for Logarithms

log 0.5
t log (0.89) Product Property for Logarithms

log 0.5

t Divide each side by log 0.89.
log 0.89
5.9480  t Use a calculator.

560 Chapter 10 Exponential and Logarithmic Relations

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 10 Resource Masters Graphing Calculator and 5-Minute Check Transparency 10-6
• Study Guide and Intervention, pp. 603–604 Spreadsheet Masters, p. 46 Answer Key Transparencies
• Skills Practice, p. 605 School-to-Career Masters, p. 20
• Practice, p. 606 Teaching Algebra With Manipulatives Technology
• Reading to Learn Mathematics, p. 607 Masters, p. 278 Interactive Chalkboard
• Enrichment, p. 608
• Assessment, p. 624
 It will take approximately 6 hours for half of the caffeine to be
eliminated from a person’s body.
Examine Use the formula to find how much of the original 130 milligrams of
2 Teach
caffeine would remain after 6 hours.
EXPONENTIAL DECAY
y
a(1  r)t Exponential decay formula
y
130(1  0.11)6 Replace a with 130, r with 0.11, and t with 6.
Teaching Tip In Example 1, point
y  64.6 Use a calculator.
out that you are calculating how
Half of 130 is 65, so the answer seems reasonable. long until half the caffeine has been
eliminated, which also means half
the caffeine remains. If the value to
Another model for exponential decay is given by y
aekt, where k is a constant.
This is the model preferred by scientists. Use this model to solve problems involving be found is something other than
radioactive decay. half, students must be careful that
they use the formula correctly.
Example 23 Exponential Decay of the Form y  aekt
PALEONTOLOGY The half-life of a radioactive substance is the time it takes In-Class Examples Power
Point®
for half of the atoms of the substance to become disintegrated. All life on Earth
contains the radioactive element Carbon-14, which decays continuously at a
fixed rate. The half-life of Carbon-14 is 5760 years. That is, every 5760 years 1 CAFFEINE Refer to Example 1.
half of a mass of Carbon-14 decays away. How long will it take for 90%
a. What is the value of k for Carbon-14? of this caffeine to be elimi-
To determine the constant k for Carbon-14, let a be the initial amount of the
nated from a person’s body?
substance. The amount y that remains after 5760 years is then represented about 20 h
1
by a or 0.5a.
2 2 GEOLOGY The half-life of
y
aekt Exponential decay formula Sodium-22 is 2.6 years.
0.5a
aek(5760) Replace y with 0.5a and t with 5760. a. What is the value of k for
0.5
e5760k Divide each side by a. Sodium-22? about 0.2666
Paleontologist ln 0.5
ln e5760k Property of Equality for Logarithmic Functions
b. A geologist examining a
Paleontologists study ln 0.5
5760k Inverse Property of Exponents and Logarithms
fossils found in geological meteorite estimates that it
ln 0.5
formations. They use 
k Divide each side by 5760. contains only about 10% as
these fossils to trace the 5760
evolution of plant and 0.00012  k Use a calculator.
much Sodium-22 as it would
animal life and the have contained when it
The constant for Carbon-14 is 0.00012. Thus, the equation for the decay of
geologic history of Earth. reached Earth’s surface. How
Carbon-14 is y
ae0.00012t, where t is given in years.
Online Research long ago did the meteorite
For information about b. A paleontologist examining the bones of a woolly mammoth estimates that reach the surface of Earth?
a career as a they contain only 3% as much Carbon-14 as they would have contained when
about 9 years ago
paleontologist, visit: the animal was alive. How long ago did the mammoth die?
www.algebra2.com/ Let a be the initial amount of Carbon-14 in the animal’s body. Then the amount
careers y that remains after t years is 3% of a or 0.03a.
Source: U.S. Department of Labor
y
ae0.00012t Formula for the decay of Carbon-14
0.03a
ae0.00012t Replace y with 0.03a.
0.03
e0.00012t Divide each side by a.
ln 0.03
ln e0.00012t Property of Equality for Logarithms
ln 0.03
0.00012t Inverse Property of Exponents and Logarithms
ln 0.03

t Divide each side by 0.00012.
0.00012
29,221  t Use a calculator.

The mammoth lived about 29,000 years ago.

www.algebra2.com/extra_examples Lesson 10-6 Exponential Growth and Decay 561

Differentiated Instruction
Logical Have students work in pairs or small groups. Ask them to
examine the growth and decay formulas used in Examples 1–4 and to
discuss how the equations are related. In particular, ask them to discuss
how they can identify which equations are used for exponential decay
situations (minus/negative sign) and which are used for exponential
growth.

Lesson 10-6 Exponential Growth and Decay 561

EXPONENTIAL GROWTH EXPONENTIAL GROWTH When a quantity increases by a fixed percent each
time period, the amount y of that quantity after t time periods is given by y
a(1  r)t,
In-Class Examples Power where a is the initial amount and r is the percent of increase expressed as a decimal.
Point® The percent of increase r is also referred to as the rate of growth .

3 The population of a city of

one million is increasing at a
Standardized Example 3 Exponential Growth of the Form y  a(1  r)t
Test Practice
rate of 3% per year. If the Multiple-Choice Test Item
population continues to grow
at this rate, in how many In 1910, the population of a city was 120,000. Since then, the population has
increased by exactly 1.5% per year. If the population continues to grow at this
years will the population rate, what will the population be in 2010?
have doubled? D
A 138,000 B 531,845
A 4 years B 5 years C 1,063,690 D 1.4 1011
C 20 years D 23 years
Read the Test Item
4 POPULATION As of 2000, You need to find the population of the city 2010  1910 or 100 years later. Since the
Nigeria had an estimated Test-Taking Tip population is growing at a fixed percent each year, use the formula y
a(1  r)t.
population of 127 million To change a percent to a Solve the Test Item
people and the United States decimal, drop the percent
y
a(1  r)t Exponential growth formula
had an estimated population symbol and move the
decimal point two places y
120,000(1  0.015)100 Replace a = 120,000, r with 0.015, and t with 2010  1910 or 100.
of 278 million people. The to the left.
populations of Nigeria and 1.5%
0.015 y
120,000(1.015)100 Simplify.

the United States can be y  531,845.48 Use a calculator.

modeled by N(t)
127e 0.026t The answer is B.
and U(t)
278e 0.009t, respec-
tively. According to these
models, when will Nigeria’s Another model for exponential growth, preferred by scientists, is y
aekt, where k
population be more than the is a constant. Use this model to find the constant k.
population of the United
States? after 46 years or in 2046 Example 4 Exponential Growth of the Form y  aekt
POPULATION As of 2000, China was the world’s most populous country, with
an estimated population of 1.26 billion people. The second most populous
country was India, with 1.01 billion. The populations of India and China can be
modeled by I(t)  1.01e0.015t and C(t)  1.26e0.009t, respectively. According to
these models, when will India’s population be more than China’s?
You want to find t such that I(t) C(t).
I(t) C(t)
1.01e0.015t 1.26e0.009t Replace I(t) with 1.01e0.015t and C(t) with 1.26e0.009t

ln 1.01e0.015t ln 1.26e0.009t Property of Inequality for Logarithms

ln 1.01  ln e0.015t ln 1.26  ln e0.009t Product Property of Logarithms

ln 1.01  0.015t ln 1.26  0.009t Inverse Property of Exponents

and Logarithms
0.006t ln 1.26  ln 1.01 Subtract 0.009 from each side.

ln 1.26  ln 1.01
t  Divide each side by 0.006.
0.006
t 36.86 Use a calculator.

After 37 years or in 2037, India will be the most populous country in the world.

562 Chapter 10 Exponential and Logarithmic Relations

Standardized Example 3 On all standardized tests, students should look to

identify any answer choices that can be logically eliminated.
Test Practice In Example 3, students can quickly determine that since 1%
of 120,000 is 1200 and therefore 1.5% is 1800, over the
100 years from 1920 to 2010 the city’s population will have increased by more than
1800(100) or 180,000 people. So Choice A is much too low. Choice D can also be
eliminated, because 1.4 1011 written in standard notation is 140,000,000,000 (which is
140 billion). That’s more than the population of the entire planet! So, the answer must be
either Choice B or Choice C.

562 Chapter 10 Exponential and Logarithmic Relations

 Concept Check 1. Write a general formula for exponential growth and decay where r is the
3 Practice/Apply
percent of change. y  a(1  r) , where r  0 represents exponential growth
t
and r  0 represents
2. Explain how to solve y
(1  r)t for t. See margin.
exponential decay
3. OPEN ENDED Give an example of a quantity that grows or decays at a
fixed rate. Sample answer: money in a bank
Study Notebook
Have students—
Guided Practice SPACE For Exercises 4–6, use the following information. • complete the definitions/examples
A radioisotope is used as a power source for a satellite. The power output P (in
25

t for the remaining terms on their
watts) is given by P
50e 0 , where t is the time in days.
Vocabulary Builder worksheets for
GUIDED PRACTICE KEY 4. Is the formula for power output an example of exponential growth or decay?
Exercises Examples Explain your reasoning. Decay; the exponent is negative. Chapter 10.
4–6 2 5. Find the power available after 100 days. about 33.5 watts • record the formulas for
7, 8 4 6. Ten watts of power are required to operate the equipment in the satellite. How exponential growth and decay.
9 1, 3 long can the satellite continue to operate? about 402 days
• include any other item(s) that they
POPULATION GROWTH For Exercises 7 and 8, use the following information. find helpful in mastering the skills
The city of Raleigh, North Carolina, grew from a population of 212,000 in 1990 to a in this lesson.
population of 259,000 in 1998.
7. Write an exponential growth equation of the form y
aekt for Raleigh, where t is
the number of years after 1990. y  212,000e0.025t
8. Use your equation to predict the population of Raleigh in 2010.
about 349,529 people
Standardized 9. Suppose the weight of a bar of soap decreases by 2.5% each time it is used. If About the Exercises…
Test Practice the bar weighs 95 grams when it is new, what is its weight to the nearest gram
after 15 uses? C Organization by Objective
A 57.5 g B 59.4 g C 65 g D 93 g • Exponential Decay: 10–20
★ indicates increased difficulty • Exponential Growth: 10–20
Practice and Apply
Assignment Guide
Homework Help 10. COMPUTERS Zeus Industries bought a computer for $2500. It is expected to
For See depreciate at a rate of 20% per year. What will the value of the computer be in Basic: 11, 13, 17, 18, 21–40
Exercises Examples 2 years? $1600
10 1
Average: 11, 13, 15–18, 21–40
12–14, 2
11. REAL ESTATE The Martins bought a condominium for $85,000. Assuming that Advanced: 10–14 even, 15, 16,
11, 17–20 3
15, 16 4
the value of the condo will appreciate at most 5% a year, how much will the 19–40
condo be worth in 5 years? at most $108,484.93
Extra Practice
See page 851. 12. MEDICINE Radioactive iodine is used to determine the health of the thyroid
gland. It decays according to the equation y
ae0.0856t, where t is in days. Find Answer
the half-life of this substance. about 8.1 days
2. Take the common logarithm of
13. PALEONTOLOGY A paleontologist finds a bone that might be a dinosaur bone.
1 each side, use the Power Property
In the laboratory, she finds that the Carbon-14 found in the bone is  of that
12 to write log (1  r)t as
found in living bone tissue. Could this bone have belonged to a dinosaur? t log (1  r), and then divide each
Explain your reasoning. (Hint: The dinosaurs lived from 220 million years ago
to 63 million years ago.) No; the bone is only about 21,000 years old, and side by the quantity log (1  r).
dinosaurs died out 63,000,000 years ago.
14. more than 44,000 14. ANTHROPOLOGY An anthropologist finds there is so little remaining
years ago Carbon-14 in a prehistoric bone that instruments cannot measure it. This means
that there is less than 0.5% of the amount of Carbon-14 the bones would have
contained when the person was alive. How long ago did the person die?
www.algebra2.com/self_check_quiz Lesson 10-6 Exponential Growth and Decay 563

Lesson 10-6 Exponential Growth and Decay 563

 Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____
BIOLOGY For Exercises 15 and 16, use the following information.
10-6 Study Guide and
p. 603 (shown) and p. 604 Bacteria usually reproduce by a process known as binary fission. In this type of
Exponential Growth and Decay
Exponential Decay Depreciation of value and radioactive decay are examples of
reproduction, one bacterium divides, forming two bacteria. Under ideal conditions,

Lesson 10-6
exponential decay. When a quantity decreases by a fixed percent each time period, the
amount of the quantity after t time periods is given by y
a(1  r) t, where a is the initial
some bacteria reproduce every 20 minutes. 15. about 0.0347
amount and r is the percent decrease expressed as a decimal.
Another exponential decay model often used by scientists is y
aekt, where k is a constant.
15. Find the constant k for this type of bacteria under ideal conditions.
Example CONSUMER PRICES As technology advances, the price of many
technological devices such as scientific calculators and camcorders goes down.
One brand of hand-held organizer sells for $89. 16. Write the equation for modeling the exponential growth of this bacterium.
y  ae0.0347t
a. If its price decreases by 6% per year, how much will it cost after 5 years?
Use the exponential decay model with initial amount $89, percent decrease 0.06, and
time 5 years.
y
a(1  r) t Exponential decay formula
y
89(1  0.06) 5
y
$65.32
a
89, r
0.06, t
5
ECONOMICS For Exercises 17 and 18, use the following information.
After 5 years the price will be $65.32. The annual Gross Domestic Product (GDP) of a country is the value of all of the
b. After how many years will its price be $50?
To find when the price will be $50, again use the exponential decay formula and solve for t.
Olympics goods and services produced in the country during a year. During the period
y
a(1  r) t
50
89(1  0.06) t
Exponential decay formula
y
50, a
89, r
0.06 1985–1999, the Gross Domestic Product of the United States grew about 3.2% per
50

(0.94) t The women’s high jump
89
Divide each side by 89.
year, measured in 1996 dollars. In 1985, the GDP was $5717 billion.
 
50
log 
log (0.94) t
89
Property of Equality for Logarithms competition first took place
 
50
log 
t log 0.94
89
Power Property in the USA in 1895, but it 17. Assuming this rate of growth continues, what will the GDP of the United States
 50 
log 
89
did not become an be in the year 2010? $12,565 billion
t
 Divide each side by log 0.94.
log 0.94
t  9.3
Olympic event until 1928.
The price will be $50 after about 9.3 years. Source: www.princeton.edu 18. In what year will the GDP reach $20 trillion? about 2025
Exercises
1. BUSINESS A furniture store is closing out its business. Each week the owner lowers
prices by 25%. After how many weeks will the sale price of a $500 item drop below $100?
6 weeks 19. OLYMPICS In 1928, when the high jump was first introduced as a women’s
CARBON DATING Use the formula y  ae0.00012t, where a is the initial amount of ★ sport at the Olympic Games, the winning women’s jump was 62.5 inches, while
Carbon-14, t is the number of years ago the animal lived, and y is the remaining
amount after t years. the winning men’s jump was 76.5 inches. Since then, the winning jump for
2. How old is a fossil remain that has lost 95% of its Carbon-14? about 25,000 years old
women has increased by about 0.38% per year, while the winning jump for men
3. How old is a skeleton that has 95% of its Carbon-14 remaining? about 427.5 years old
has increased at a slower rate, 0.3%. If these rates continue, when will the
Gl

Skills
NAME
/M G ______________________________________________
Hill

Practice, p. 605 and

603 DATE ____________
GlPERIOD
Al _____
b 2
women’s winning high jump be higher than the men’s? after the year 2182
10-6 Practice (Average)
Practice, p.Growth
Exponential 606and(shown)
Decay

★ 20. HOME OWNERSHIP The Mendes family bought a new house 10 years ago for
Solve each problem.

1. INVESTING The formula A

P 1    2 
r 2t
gives the value of an investment after t years in
an account that earns an annual interest rate r compounded twice a year. Suppose $500
is invested at 6% annual interest compounded twice a year. In how many years will the
$120,000. The house is now worth $191,000. Assuming a steady rate of growth,
investment be worth $1000? about 11.7 yr
what was the yearly rate of appreciation? 4.7%
2. BACTERIA How many hours will it take a culture of bacteria to increase from 20 to
2000 if the growth rate per hour is 85%? about 7.5 h

3. RADIOACTIVE DECAY A radioactive substance has a half-life of 32 years. Find the

constant k in the decay formula for the substance. about 0.02166 21. CRITICAL THINKING The half-life of Radium is 1620 years. When will a
4. DEPRECIATION A piece of machinery valued at $250,000 depreciates at a fixed rate of
20-gram sample of Radium be completely gone? Explain your reasoning.
12% per year. After how many years will the value have depreciated to $100,000?
about 7.2 yr
Never; theoretically, the amount left will always be half of the previous
5. INFLATION For Dave to buy a new car comparably equipped to the one he bought 8 years amount.
ago would cost $12,500. Since Dave bought the car, the inflation rate for cars like his has
been at an average annual rate of 5.1%. If Dave originally paid $8400 for the car, how 22. WRITING IN MATH Answer the question that was posed at the beginning of
long ago did he buy it? about 8 yr
the lesson. See margin.
6. RADIOACTIVE DECAY Cobalt, an element used to make alloys, has several isotopes.
One of these, cobalt-60, is radioactive and has a half-life of 5.7 years. Cobalt-60 is used to
trace the path of nonradioactive substances in a system. What is the value of k for
How can you determine the current value of your car?
Cobalt-60? about 0.1216
Include the following in your answer:
7. WHALES Modern whales appeared 510 million years ago. The vertebrae of a whale
discovered by paleontologists contain roughly 0.25% as much carbon-14 as they would
have contained when the whale was alive. How long ago did the whale die? Use
• a description of how to find the percent decrease in the value of the car each
k
0.00012. about 50,000 yr
year, and
8. POPULATION The population of rabbits in an area is modeled by the growth equation
P(t)
8e0.26t, where P is in thousands and t is in years. How long will it take for the
• a description of how to find the value of a car for any given year when the
population to reach 25,000? about 4.4 yr
rate of depreciation is known.
9. DEPRECIATION A computer system depreciates at an average rate of 4% per month. If
the value of the computer system was originally $12,000, in how many months is it
worth $7350? about 12 mo

10. BIOLOGY In a laboratory, a culture increases from 30 to 195 organisms in 5 hours.

What is the hourly growth rate in the growth formula y
a(1  r) t ? about 45.4%
Standardized 23. SHORT RESPONSE An artist creates a sculpture out of salt that weighs
Test Practice 2000 pounds. If the sculpture loses 3.5% of its mass each year to erosion,
Gl
Reading
NAME
/M G ______________________________________________
Hill 606 DATE ____________
Gl PERIOD
Al _____
b 2
after how many years will the statue weigh less than 1000 pounds? about 19.5 yr
10-6 Readingto
to Learn
Learn Mathematics ELL
Mathematics, p. 607
Exponential Growth and Decay 24. The curve shown at the right represents a y
Pre-Activity How can you determine the current value of your car?
Read the introduction to Lesson 10-6 at the top of page 560 in your textbook.
portion of the graph of which function? D
• Between which two years shown in the table did the car depreciate by
the greatest amount?
A y
50  x B y
log x
between years 0 and 1
• Describe two ways to calculate the value of the car 6 years after it was C y
ex D xy
5
purchased. (Do not actually calculate the value.)
Sample answer: 1. Multiply $9200.66 by 0.16 and subtract the
result from $9200.66. 2. Multiply $9200.66 by 0.84.

O x
Reading the Lesson
1. State whether each situation is an example of exponential growth or decay.

a. A city had 42,000 residents in 1980 and 128,000 residents in 2000. growth
564 Chapter 10 Exponential and Logarithmic Relations
b. Raul compared the value of his car when he bought it new to the value when he
traded ‘;lpit in six years later. decay

c. A paleontologist compared the amount of carbon-14 in the skeleton of an animal

when it died to the amount 300 years later. decay

d. Maria deposited $750 in a savings account paying 4.5% annual interest compounded NAME ______________________________________________ DATE ____________ PERIOD _____
quarterly. She did not make any withdrawals or further deposits. She compared the
balance in her passbook immediately after she opened the account to the balance
3 years later. growth Enrichment,
10-6 Enrichment p. 608
2. State whether each equation represents exponential growth or decay.

a. y
5e0.15t growth b. y
1000(1  0.05) t decay
Effective Annual Yield
When interest is compounded more than once per year, the effective annual
c. y
0.3e1200t decay d. y
2(1  0.0001) t growth
yield is higher than the annual interest rate. The effective annual yield, E, is
the interest rate that would give the same amount of interest if the interest
were compounded once per year. If P dollars are invested for one year, the
Helping You Remember value of the investment at the end of the year is A
P(1  E). If P dollars
are invested for one year at a nominal rate r compounded n times per year,
3. Visualizing their graphs is often a good way to remember the difference between
mathematical equations. How can your knowledge of the graphs of exponential equations  
r n
the value of the investment at the end of the year is A
P 1   . Setting
n
from Lesson 10-1 help you to remember that equations of the form y
a(1  r) t the amounts equal and solving for E will produce a formula for the effective
represent exponential growth, while equations of the form y
a(1  r) t represent annual yield.
exponential decay?
 nr 
n
Sample answer: If a  0, the graph of y  ab x is always increasing if P(1  E)
P 1  
b  1 and is always decreasing if 0  b  1. Since r is always a positive
 nr 
n
number, if b  1  r, the base will be greater than 1 and the function will 1  E
1  
be increasing (growth), while if b  1  r, the base will be less than 1
E
1    1
and the function will be decreasing (decay). r n
n

If compounding is continuous, the value of the investment at the end of one

year is A
Pe r Again set the amounts equal and solve for E A formula for

564 Chapter 10 Exponential and Logarithmic Relations

Maintain Your Skills
Mixed Review Write an equivalent exponential or logarithmic equation. (Lesson 10-5)
4 Assess
25. e3
y ln y  3 26. e4n  2
29 27. ln 4  2 ln x
8 Open-Ended Assessment
ln 29  4n  2 4x2  e8
Solve each equation or inequality. Round to four decimal places. (Lesson 10-4) Modeling Using manipulatives,
28. 16x
70 1.5323 29. 23p 1000 p  3.3219 30. logb 81
2 9 ask students to demonstrate why
the amount of compound interest
BUSINESS For Exercises 31–33, use the following information. earned annually increases each
The board of a small corporation decided that 8% of the annual profits would be
divided among the six managers of the corporation. There are two sales managers
year. Have students relate this to
and four nonsales managers. Fifty percent of the amount would be split equally their understanding of
among all six managers. The other 50% would be split among the four nonsales exponential growth.
managers. Let p represent the annual profits of the corporation. (Lesson 9-2)
31. Write an expression to represent the share of the profits each nonsales manager Assessment Options
will receive. 0.5(0.08p) 0.5(0.08p)
p    Quiz (Lesson 10-6) is available
32. Simplify this expression.  6 4
60 on p. 624 of the Chapter 10
33. Write an expression in simplest form to represent the share of the profits each
sales manager will receive. p Resource Masters.
150
Without writing the equation in standard form, state whether the graph of each
equation is a parabola, circle, ellipse, or hyperbola. (Lesson 8-6) Answer
34. 4y2  3x2  8y  24x
50 hyperbola 35. 7x2  42x  6y2  24y
45 ellipse
22. Answers should include the
36. y2  3x  8y
4 parabola 37. x2  y2  6x  2y  5
0 circle
following.
AGRICULTURE For Exercises 38–40, • Find the absolute value of the
use the graph at the right. (Lesson 5-1)
USA TODAY Snapshots® difference between the price of
38. Write the number of pounds the car for two consecutive years.
of pecans produced by U.S. Georgia led pecan production in 2000
growers in 2000 in scientific U.S. growers produced more than 206 million pounds Then divide this difference by
of pecans in 2000. States producing the most pecans
notation. 2.06  108 (in pounds): the price of the car for the
39. Write the number of pounds of earlier year.
pecans produced by the state of Georgia 80 million • Find 1 minus the rate of decrease
Georgia in 2000 in scientific
notation. 8  107 New Mexico 32 million in the value of the car as a deci-
mal. Raise this value to the num-
40. What percent of the overall pecan Texas 30 million
production for 2000 can be ber of years it has been since
Louisiana 17 million the car was purchased, and then
attributed to Georgia? about
38.8% Alabama multiply by the original value of
15 million
the car.
Arizona 14 million

Source: National Agricultural Statistics Service

By Sam Ward, USA TODAY

On Quake Anniversary, Japan Still Worries

It is time to complete your project. Use the information and data
you have gathered about earthquakes to prepare a research report
or Web page. Be sure to include graphs, tables, diagrams, and
any calculations you need for the earthquake you chose.
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Lesson 10-6 Exponential Growth and Decay 565

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Lesson 10-6 Exponential Growth and Decay 565

 Study Guide
and Review
Vocabulary and Concept Check
Vocabulary and
Change of Base Formula (p. 548) natural base, e (p. 554) Property of Equality for
Concept Check common logarithm (p. 547) natural base exponential function Logarithmic Functions (p. 534)
exponential decay (p. 524) (p. 554) Property of Inequality for
• This alphabetical list of exponential equation (p. 526) natural logarithm (p. 554) Exponential Functions (p. 527)
vocabulary terms in Chapter exponential function (p. 524) natural logarithmic function (p. 554) Property of Inequality for
10 includes a page reference exponential growth (p. 524) Power Property of Logarithms Logarithmic Functions (p. 534)
where each term was exponential inequality (p. 527) (p. 543) Quotient Property of Logarithms
logarithm (p. 531) Product Property of Logarithms (p. 542)
introduced. rate of decay (p. 560)
logarithmic equation (p. 533) (p. 541)
• Assessment A vocabulary logarithmic function (p. 532) Property of Equality for Exponential rate of growth (p. 562)
test/review for Chapter 10 is logarithmic inequality (p. 533) Functions (p. 526)
available on p. 622 of the
Chapter 10 Resource Masters.
State whether each sentence is true or false. If false, replace the underlined
word(s) to make a true statement.
Lesson-by-Lesson 1. If 242y  3
24y  4, then 2y  3
y  4 by the Property of Equality for 3. false; common
Exponential Functions . true logarithm
Review
2. The number of bacteria in a petri dish over time is an example of exponential 4. false; Property
decay. false; exponential growth of Inequality for
For each lesson, Logarithms
3. The natural logarithm is the inverse of the exponential function with base 10.
• the main ideas are 5. true
summarized, 4. The Power Property of Logarithms shows that ln 9  ln 81. 7. false;
5. If a savings account yields 2% interest per year, then 2% is the rate of growth. logarithmic
• additional examples review function
concepts, and 6. Radioactive half-life is an example of exponential decay. true
8. false; Product
• practice exercises are provided. 7. The inverse of an exponential function is a composite function. Property of
8. The Quotient Property of Logarithms is shown by log4 2x
log4 2  log4 x.
Logarithms

9. The function f(x)

2(5)x is an example of a quadratic function. false; exponential function
Vocabulary
PuzzleMaker
ELL The Vocabulary PuzzleMaker
software improves students’ mathematics
vocabulary using four puzzle formats—
crossword, scramble, word search using a 10-1 Exponential Functions
word list, and word search using clues. See pages Concept Summary
Students can work on a computer screen 523–530.
or from a printed handout. • An exponential function is in the form y
abx, where a ≠ 0,
b 0, and b  1.
• The function y
abx represents exponential growth for a 0 and b 1,
MindJogger and exponential decay for a 0 and 0  b  1.
Videoquizzes • Property of Equality for Exponential Functions:
If b is a positive number other than 1, then bx
by if and only if x
y.
ELL MindJogger Videoquizzes • Property of Inequality for Exponential Functions:
provide an alternative review of concepts If b 1, then bx by if and only if x y, and bx  by if and only if x  y.
presented in this chapter. Students work
in teams in a game show format to gain 566 Chapter 10 Exponential and Logarithmic Relations www.algebra2.com/vocabulary_review
points for correct answers. The questions
are presented in three rounds.
Round 1 Concepts (5 questions) TM

Round 2 Skills (4 questions) Have students look through the chapter to make sure they have
Round 3 Problem Solving (4 questions) included notes and examples for each lesson in this chapter in
their Foldable.
For more information
Encourage students to refer to their Foldables while completing
about Foldables, see
the Study Guide and Review and to use them in preparing for the
Teaching Mathematics
Chapter Test.
with Foldables.

566 Chapter 10 Exponential and Logarithmic Relations

 Chapter 10 Study Guide and Review Study Guide and Review

Example Solve 64  23n  1 for n.

64
23n  1 Original equation

26
23n  1 Rewrite 64 as 26 so each side has the same base.

6
3n  1 Property of Equality for Exponential Functions
5 5

n The solution is .
3 3

Exercises Determine whether each function represents exponential growth or

decay. See Example 2 on page 525.
1
10. y
5(0.7)x decay 11. y
(4)x growth
3
Write an exponential function whose graph passes through the given points.
See Example 3 on page 525.
1 x
12. (0, 2) and (3, 54) y  2(3)x 13. (0, 7) and (1, 1.4) y  7 
5  
Solve each equation or inequality. See Examples 5 and 6 on pages 526 and 527.
1
14. 9x
 2 15. 26x
45x  2 1
81
7
x
6 or x  6
2 2
16. 493p  1
72p  5  17. 9x  27x 2
4

10-2 Logarithms and Logarithmic Functions

See pages
Concept Summary Examples
531–538.
• Suppose b 0 and b ≠ 1. For x 0, there is a log7 x
2 → 72
x
number y such that logb x
y if and only if by
x.
• Logarithmic to exponential inequality: log2 x 5 → x 25
If b 1, x 0, and logb x y, then x by. log3 x  4 → 0  x  34
If b 1, x 0, and logb x  y, then 0  x  by.

• Property of Equality for Logarithmic Functions: If log5 x

log5 6,
If b is a positive number other than 1, then x
6.
then logb x
logb y if and only if x
y.
• Property of Inequality for Logarithmic Functions: If log4 x log4 10,
If b 1, then logb x logb y if and only if x y, then x 10.
and logb x  logb y if and only if x  y.

3
Examples 1 Solve log9 n  .
2
3
log9 n  Original inequality
2
3
n 92 Logarithmic to exponential inequality

 3
n (32) 2 9
32
n 33 Power of a Power
n 27 Simplify.

Chapter 10 Study Guide and Review 567

Chapter 10 Study Guide and Review 567

Study Guide and Review Chapter 10 Study Guide and Review

2 Solve log3 12  log3 2x.

log3 12
log3 2x Original equation
12
2x Property of Equality for Logarithmic Functions
6
x Divide each side by 2.

Exercises Write each equation in logarithmic form. See Example 1 on page 532.
1 1 3 3
18. 73
343 log7 343  3 19. 52
 log5   2 20. 4 2
8 log4 8  
25 25 2
1
Write each equation in exponential form. See Example 2 on page 532. 23. 62  
1 36
1  1
21. log4 64
3 43  64 22. log8 2
 8 3  2 23. log6 
2
3 36
Evaluate each expression. See Examples 3 and 4 on pages 532 and 533.
1
24. 4log4 9 9 25. log7 75 5 26. log81 3  27. log13 169 2
4
Solve each equation or inequality. See Examples 5–8 on pages 533 and 534.
1 3
28. log4 x
 2 29. log81 729
x 
2 2 1
30. logb 9
2 3 31. log8 (3y  1)  log8 (y  5) 3  y  3
32. log5 12  log5 (5x  3) x  3 33. log8 (x2  x)
log8 12 4, 3

10-3 Properties of Logarithms

See pages Concept Summary
541–546.
• The logarithm of a product is the sum of the logarithms of its factors.
• The logarithm of a quotient is the difference of the logarithms of the
numerator and the denominator.
• The logarithm of a power is the product of the logarithm and the exponent.
Example Use log12 9  0.884 and log12 18  1.163 to approximate the value of log12 2.
18 18
log12 2
log12  Replace 2 with .
9
9

log12 18  log12 9 Quotient Property

 1.163  0.884 or 0.279 Replace log12 9 with 0.884 and log12 18 with 1.163.

Exercises Use log9 7  0.8856 and log9 4  0.6309 to approximate

the value of each expression. See Examples 1 and 2 on page 542.
34. log9 28 1.5165 35. log9 49 1.7712 36. log9 144 2.2618

Solve each equation. See Example 5 on page 543.

1 1
37. log2 y
 log2 27 3 38. log5 7   log5 4
log5 x 14
3 2
39. 2 log2 x  log2 (x  3)
2 6 40. log3 x  log3 4
log3 12 48
16
41. log6 48  log6   log6 5
log6 5x
5
15 42. log7 m
13 log7 64  12 log7 121 44

568 Chapter 10 Exponential and Logarithmic Relations

568 Chapter 10 Exponential and Logarithmic Relations

 Chapter 10 Study Guide and Review Study Guide and Review

10-4 Common Logarithms

See pages
Concept Summary
547–551.
• Base 10 logarithms are called common logarithms and are usually written
without the subscript 10: log10 x
log x.
• You use the inverse of logarithms, or exponentiation, to solve equations or
inequalities involving common logarithms: 10log x
x.
log n ← log base b original number
• The Change of Base Formula: loga n
 
b
log a ← log base b old base
b

Example Solve 5x  7.
5x
7 Original equation

log 5x
log 7 Property of Equality for Logarithmic Functions

x log 5
log 7 Power Property of Logarithms

log 7
x
 Divide each side by log 5.
log 5
0.8451
x   or 1.2090 Use a calculator.
0.6990

Exercises Solve each equation or inequality. Round to four decimal places.

See Examples 3 and 4 on page 548.
2
43. 2x
53 5.7279 44. 2.3x
66.6 2.2452 45. 34x  7  42x  3 x  7.3059
46. 63y
8y  1 0.6309 47. 12x  5  9.32 48. 2.1x  5
9.32 8.0086
x  5.8983
Express each logarithm in terms of common logarithms. Then approximate its
value to four decimal places. See Example 5 on page 549.
log 11 log 15 log 1000
49. log4 11 ; 1.7297 50. log2 15 ; 3.9069 51. log20 1000 ; 2.3059
log 4 log 2 log 20

10-5 Base e and Natural Logarithms

See pages Concept Summary
554–559.
• You can write an equivalent base e exponential equation for a natural
logarithmic equation and vice versa by using the fact that ln x
loge x.
• Since the natural base function and the natural logarithmic function are
inverses, these two functions can be used to “undo” each other.
eln x
x and ln ex
x

Example Solve ln (x  4)  5.
ln (x  4) 5 Original inequality

eln (x  4) e5 Write each side using exponents and base e.

x4 e5 Inverse Property of Exponents and Logarithms

x e5 4 Subtract 4 from each side.

x 144.4132 Use a calculator.

Chapter 10 Study Guide and Review 569

Chapter 10 Study Guide and Review 569

 • Extra Practice, see pages 849–851.
Study Guide and Review • Mixed Problem Solving, see page 871.

Exercises Write an equivalent exponential or logarithmic equation.

See Example 3 on page 555.
52. ex
6 ln 6  x 53. ln 7.4
x ex  7.4

Evaluate each expression. See Example 4 on page 555.

54. eln 12 12 55. ln e7x 7x

Solve each equation or inequality.

See Examples 5 and 7 on pages 555 and 556.
56. 2ex  4
1 0.9163 57. ex 3.2 x  1.1632 58. 4e2x  15
7 0.3466
59. ln 3x  5 60. ln (x  10)
0.5 61. ln x  ln 4x
10
0  x
49.4711 11.6487 74.2066

10-6 Exponential Growth and Decay

See pages
Concept Summary
560–565.
• Exponential decay: y
a(1  r)t or y
aekt
• Exponential growth: y
a(1  r)t or y
aekt
Example BIOLOGY A certain culture of bacteria will grow from 500 to 4000 bacteria in
1.5 hours. Find the constant k for the growth formula. Use y = nekt.
y
aekt Exponential growth formula

4000
500ek(1.5) Replace y with 4000, a with 500, and t with 1.5.

8
e1.5k Divide each side by 500.

ln 8
ln e1.5k Property of Equality for Logarithmic Functions

ln 8
1.5k Inverse Property of Exponents and Logarithms

ln 8

k Divide each side by 1.5.
1.5
1.3863  k Use a calculator.

Exercises See Examples 1–4 on pages 560–562.

62. BUSINESS Able Industries bought a fax machine for $250. It is expected to
depreciate at a rate of 25% per year. What will be the value of the fax machine in
3 years? $105.47

63. BIOLOGY For a certain strain of bacteria, k is 0.872 when t is measured in days.
How long will it take 9 bacteria to increase to 738 bacteria? 5.05 days

64. CHEMISTRY Radium-226 decomposes radioactively. Its half-life, the time it takes
for half of the sample to decompose, is 1800 years. Find the constant k in the decay
formula for this compound. about –0.000385

65. POPULATION The population of a city 10 years ago was 45,600. Since then, the
population has increased at a steady rate each year. If the population is currently
64,800, find the annual rate of growth for this city. about 3.6%

570 Chapter 10 Exponential and Logarithmic Relations

570 Chapter 10 Exponential and Logarithmic Relations

 Practice Test

Vocabulary and Concepts

Assessment Options
Choose the term that best completes each sentence.
Vocabulary Test A vocabulary
1. The equation y
0.3(4)x is an exponential ( growth, decay) function.
test/review for Chapter 10 can
2. The logarithm of a quotient is the (sum, difference ) of the logarithms of
be found on p. 622 of the
the numerator and the denominator.
Chapter 10 Resource Masters.
3. The base of a natural logarithm is (10, e ) .
Chapter Tests There are six
Skills and Applications Chapter 10 Tests and an Open-
Ended Assessment task available
4. Write 37
2187 in logarithmic form. log3 2187  7 in the Chapter 10 Resource
4 4
5. Write log8 16
 in exponential form. 83  16 Masters.
3
6. Write an exponential function whose graph passes through (0, 0.4) and (2, 6.4). y  0.4(4)x
log 5
Chapter 10 Tests
7. Express log3 5 in terms of common logarithms.  Form Type Level Pages
log 3
1
8. Evaluate log2 . 5 1 MC basic 609–610
32
2A MC average 611–612
Use log4 7  1.4037 and log4 3  0.7925 to approximate the value of each expression.
7 2B MC average 613–614
9. log4 21 2.1962 10. log4  0.3888
12 2C FR average 615–616
Simplify each expression. 2D FR average 617–618
11. 382 81 12. 815  35 335 3 FR advanced 619–620
Solve each equation or inequality. Round to four decimal places if necessary. 17. 108 19. 2, 6 22. 15 MC = multiple-choice questions
1
13. 2x  3
 1 14. 272p  1
34p  1 2 15. log2 x  7 0  x  128 FR = free-response questions
16
1
16. logm 144
2  17. log3 x  2 log3 2
3 log3 3 18. log9 (x  4)  log9 (x  4)
1 5 Open-Ended Assessment
12
19. log5 (8y  7)
log5 (y  5) 20. log3 3(4x  1)
15 4
2 21. 7.6x  1
431 3.9910 Performance tasks for Chapter 10
1
22. log2 5   log2 27
log2 x 23. 3x
5x  1 3.1507 24. 42x  3
9x  3 18.6848 can be found on p. 621 of the
3
25. e3y 6 y  0.5973 26. 2e3x  5
11 0.3662 27. ln 3x  ln 15
2 36.9453 Chapter 10 Resource Masters. A
sample scoring rubric for these
COINS For Exercises 28 and 29, use the following information. tasks appears on p. A25.
You buy a commemorative coin for $25. The value of the coin increases 3.25% per year.
28. How much will the coin be worth in 15 years? $40.39 Unit 3 Test A unit test/review
29. After how many years will the coin have doubled in value? 22 can be found on pp. 629–630 of
the Chapter 10 Resource Masters.
30. QUANTITATIVE COMPARISION Compare Column A Column B
the quantity in Column A and the quantity
in Column B. Then determine whether: B $100 was deposited in an account 5 years ago.
TestCheck and
A the quantity in Column A is greater,
the current value of the current value of Worksheet Builder
B the quantity in Column B is greater, the account if the the account if the This networkable software has
annual interest rate annual interest rate is
C the two quantities are equal, or
is 3% compunded 3% compounded three modules for assessment.
D the relationship cannot be determined quarterly continuously • Worksheet Builder to make
from the information given. worksheets and tests.
www.algebra2.com/chapter_test Chapter 10 Practice Test 571
• Student Module to take tests
on-screen.
• Management System to keep
Portfolio Suggestion student records.
Introduction In mathematics, exponential functions can be used to model
real-world problems. The solution to the exponential function provides a
solution to the real-world problem.
Ask Students Find a real-world problem modeled by an exponential function
from your work in this chapter and show how you solved it. Explain how the
function models the real-world situation and what could be gained by
understanding the real-world problem better. Place your work in your portfolio.

Chapter 10 Practice Test 571

 Standardized
Test Practice

These two pages contain practice (xy)2z0

Part 1 Multiple Choice 3
D
5. 2
yx
questions in the various formats
that can be found on the most Record your answers on the answer sheet 1 z
A 2 B 
provided by your teacher or on a sheet of xy x2
frequently given standardized paper. z 1
tests. 1. The arc shown is y
C 
x
D 
x
part of a circle. Find 8
A practice answer sheet for these the area of the v2  36
shaded region. B 6. If 
10, then v
A
two pages can be found on p. A1 4 6v
of the Chapter 10 Resource A 8 units2 A 16. B 4.
4 O 4 8 x
Masters. B 16 units2 C 4. D 8.
4
NAME

Standardized
DATE PERIOD
C 32 units2
10 Standardized Test Practice
Test Practice 1
Student Recording
Student Record Sheet,
Sheet (Use with pages 572–573 of p. A1Edition.)
the Student D 64 units2  is equivalent to A
7. The expression 45
3
Part 1 Multiple Choice
Select the best answer from the choices given and fill in the corresponding oval.
2. If line ᐉ is parallel to line m in the figure below,
A 5. B .
35
1 A B C D 4 A B C D 7 A B C D 9 A B C D

2 A B C D 5 A B C D 8 A B C D 10 A B C D
what is the value of x? D C 5. D 15.
3 A B C D 6 A B C D

130˚
ᐉ
Part 2 Short Response/Grid In 8. What are all the values for x such that
150˚
Solve the problem and write your answer in the blank.
For Questions 12–18, also enter your answer by writing each number or symbol in
m x2  3x  18? B
a box. Then fill in the corresponding oval for that number or symbol.

11 13 15 17 A x  3 B 3  x  6
Answers

x˚
.
/
.
0
/
.
0
.
0
.
/
.
0
/
.
0
.
0
.
/
.
0
/
.
0
.
0
C x 3 D x6
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 5 5
A 40 B 50
9. If f(x)
2x3  18x, what are all the values
6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8

of x at which f(x)
0? B
9 9 9 9 9 9 9 9 9 9 9 9

12 14 16 18
C 60 D 70
/ / / / / / / /
A 0, 3 B 3, 0, 3
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
3. According to the graph, what was the percent
of increase in sales from 1998 to 2000? D 6, 0, 6 3, 2, 3
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
C D
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
20
Sales ($ thousands)

Part 3 Quantitative Comparison 10. Which of the following is equal to

Select the best answer from the choices given and fill in the corresponding oval. 15 17.5(102)
 ? D
19 A B C D 21 A B C D 23 A B C D

10
500(104)
0.035(102) 0.35(102)
20 A B C D 22 A B C D

A B
5
C 0.0035(102) D 0.035(102)
0
1998 1999 2000 2001
Additional Practice Year

See pp. 627–628 in the Chapter 10

A 5% B 15% C 25% D 50%
Resource Masters for additional
Test-Taking Tip
standardized test practice. 4. What is the x-intercept of the line described by Question 7 You can use estimates to help you
the equation y
2x  5? B eliminate answer choices. For example, in Question
1 1
A 5 B 
5 7, you can estimate that 45 is less than 49,
2 3 3
7 1
which is  or 2. Eliminate choices C and D.
5 3 3
C 0 D 
2
572 Chapter 10 Standardized Test Practice

TestCheck and
Log On for Test Practice
Worksheet Builder
The Princeton Review offers
additional test-taking tips and Special banks of standardized test
practice problems at their web site. Visit questions similar to those on the SAT,
www.princetonreview.com or ACT, TIMSS 8, NAEP 8, and Algebra 1
www.review.com End-of-Course tests can be found on
this CD-ROM.

572 Chapter 10 Exponential and Logarithmic Relations

 Aligned and
verified by

Part 2 Short Response/Grid In Part 3 Quantitative Comparison

Record your answers on the answer sheet Compare the quantity in Column A and the
provided by your teacher or on a sheet of quantity in Column B. Then determine
paper. whether:
11. If the outer diameter of a cylindrical tank is A the quantity in Column A is greater,
62.46 centimeters and the inner diameter is
B the quantity in Column B is greater,
53.32 centimeters, what is the thickness of
the tank? 4.57 cm C the two quantities are equal, or

62.46 cm
D the relationship cannot be determined
from the information given.

53.32 cm Column A Column B

19. 1  xy  0

Top View
D xy xy

12. What number added to 80% of itself is equal 20.

z˚
to 45? 25 r
s
C
13. Of 200 families surveyed, 95% have at least y˚ x˚
one TV and 60% of those with TVs have more t
than 2 TVs. If 50 families have exactly 2 TVs,
how many families have exactly 1 TV? 26 z
xy

14. In the figure, if ED

8, B t r
2  s2
what is the measure
15˚ 15˚
of line segment AE? 2
30˚ 21. B C
12 12
A
O

A E D C A D

circumference of circle O
8
15. If a ↔ b is defined as a – b  ab, find the
value of 4 ↔ 2. 10 perimeter of
16
square ABCD
16. If 6(m  k)
26  4(m  k), what is the
value of m  k? 13
22. xyz
5
17. If y
1  x2 and 3  x  1, what number xyz
9
is found by subtracting the least possible
value of y from the greatest possible value A xz 6
of y? 9

18. If f(x)
(x  )(x  3)(x  e), what is the 23. nx  0
difference between the greatest and least roots
of f(x)? Round to the nearest hundredth. .42 B 2nx (x  n)2

www.algebra2.com/standardized_test Chapter 10 Standardized Test Practice 573

Chapter 10 Standardized Test Practice 573

 Page 521, Chapter 10 Getting Started 4b. As the value of x increases,
9. 10. the value of y for the graph
f (x ) f (x ) x

f (x )  x  2
1
of y
 13  decreases faster
f (x )  2x 3
than for the graph of
x

O x O x
 12 
y
 , and the value of y
x

f (x )   1 x
1
for the graph of y
  14  [5, 5] scl: 1 by [2, 8] scl: 1
2
f (x )  3x  2 decreases faster than for the
x
1
graph of y
 . The graphs have the same domain,
3
11. 12. all real numbers, and range, y 0. They have the same
f (x )
asymptote, the x-axis, and the same y-intercept, 1.
1 f (x )
f (x )  x  1
4c. The graph of y
3(2)x moves
down and to the right more
f 1(x )  3x  4 quickly than the graph of
O x
y
1(2)x. The graph of
O x
y
3(2)x moves up and to
f (x )  x  1 the right more quickly than
x4
f (x ) 
3 the graph of y
2x. All of
[5, 5] scl: 1 by [5, 5] scl: 1
the graphs have the same
domain, all real numbers, and asymptote, the x-axis,
but the range of y
3(2)x and y
1(2)x is y  0,
Additional Answers for Chapter 10

Page 522, Preview of Lesson 10-1 while the range of y

2x and y
3(2)x is y 0. The
Algebra Activity y-intercept of y
3(2)x is 3, of y
1(2)x is 1, of
3. Sample graph: y y
2x is 1, and of y
3(2)x is 3.
140
Pages 528–530, Lesson 10-1
120 21. D
{x |x is all 22. D
{x |x is all
real numbers.}, real numbers.},
100
R
{y |y 0} R
{y |y 0}
80 y y

60
y  5(2)x
40
y  2(3)x
20

x O x O x
O 2 4 6

23. D
{x |x is all 24. D
{x |x is all
real numbers.}, real numbers.},
Page 524, Lesson 10-1
R
{y |y 0} R
{y |y 0}
Graphing Calculator Investigation
y y
4a. As the value of x increases,
the value of y for the graph
of y
4x increases faster y4( 13 )x
than for the graph of y
3x, y  0.5(4 )x
and the value of y for the
graph of y
3x increases
faster than for the graph of O x O x
[5, 5] scl: 1 by [2, 8] scl: 1
y
2x. The graphs have the
same domain, all real numbers, and range, y 0.
They have the same asymptote, the x-axis, and the
same y-intercept, 1.

573A Chapter 10 Additional Answers

25. D
{x |x is all 26. D
{x |x is all 73. The graphs have the same
real numbers.}, real numbers.}, shape. The graph of
x2
R
{y |y  0}
y
R
{y |y  0}
y
 15  is the graph of
y

1 x
x y
 translated two units
x
5
O O
to the right. The asymptote
x
y  2.5(5)x
[5, 5] scl: 1 by [1, 9] scl: 1
for the graph of y
 and  51 
x2
y ( 15 )x for y
 15  is the line y
0. The graphs have the
same domain, all real numbers, and range, y 0. The
x
y-intercept of the graph of y
  51  is 1 and for the
1 x2
68. The number of teams y that could compete in a graph of y

5  
is 25.
tournament in which x rounds are played can be
74. The graphs have the same
expressed as y
2x. The 2 teams that make it to the
shape. The graph of
final round got there as a result of winning games x
played with 2 other teams, for a total of 2  2
22 or 4  14   1 is the graph of
y

1 x
y
 translated one unit
games played in the previous round or semifinal
round. Answers should include the following. 4
down. The asymptote for
• Rewrite 128 as a power of 2, 27. Substitute 27 for y in x
the equation y
2x. Then, using the Property of Equal- [5, 5] scl: 1 by [3, 7] scl: 1 the graph of y
  14  is the

Additional Answers for Chapter 10

x
ity for Exponents, x must be 7. Therefore, 128 teams
would need to play 7 rounds of tournament play.
line y
0 and for the graph of y
  14   1 is the line
y
1. The graphs have the same domain, all real
• Sample answer: 52 would be an inappropriate number x
of teams to play in this type of tournament because numbers, but the range of y
  14  is y 0 and of
1 x
52 is not a power of 2.
 
y
  1 is y 1. The y-intercept of the graph of
4
71. The graphs have the same 1 x 1 x
shape. The graph of 4 
y
 is 1 and for the graph of y
  1 is 0.
4  
y
2x  3 is the graph of
y
2x translated three units Page 537, Lesson 10-2
up. The asymptote for the 66a. 66b. The graphs are
y
graph of y
2x is the line reflections of each
y
0 and for y
2x  3 is other over the line
[5, 5] scl: 1 by [1, 9] scl: 1
the line y
3. The graphs y ( 12 )x y
x.
have the same domain, all real numbers, but the range
of y
2x is y 0 and the range of y
2x  3 is y 3.
O x
The y-intercept of the graph of y
2x is 1 and for the
graph of y
2x  3 is 4. y  log 1 x
2

72. The graphs have the same

shape. The graph of
y
3x  1 is the graph of 67a. y y  log2(x  2)
y
3x translated one unit to y  log2x  3
the left. The asymptote for
the graph of y
3x and for y  log2(x  1)
y
3x  1 is the line y
0. O x
[5, 5] scl: 1 by [1, 9] scl: 1
The graphs have the same
domain, all real numbers, and range, y 0. The
y-intercept of the graph of y
3x is 1 and for the graph y  log2x  4
of y
3x  1 is 3.
67b. The graph of y
log2 x  3 is the graph of y
log2 x
translated 3 units up. The graph of y
log2 x  4 is
the graph of y
log2 x translated 4 units down. The
graph of log2 (x  1) is the graph of y
log2 x
translated 1 unit to the right. The graph of log2 (x  2)
is the graph of y
log2 x translated 2 units to the left.

Chapter 10 Additional Answers 573B

 73. A logarithmic scale illustrates that values next to each Pages 545–546, Lesson 10-3
other vary by a factor of 10. Answers should include 36. n logb x  m logb x  (n  m) logb x
the following.
logb xn  logb xm  (n  m) logb x Power Property of
• Pin drop: 1 100; Whisper: 1 102; Normal Logarithms
conversation: 1 106; Kitchen noise: 1 1010; Jet logb (xn  xm)  (n  m) logb x Product Property
engine: 1 1012 of Logarithms
• Pin Whisper Normal Kitchen Jet logb (xn  m)  (n  m) logb x Product of Powers
drop (4 feet) conversation noise engine Property
(n  m) logb x
(n  m) logb x ✓ Power Property of
Logarithms
0 2  10 11 4  10 11 6  10 11 8  10 11 1  10 12

• On the scale shown above, the sound of a pin drop

and the sound of normal conversation appear not to
differ by much at all, when in fact they do differ in
terms of the loudness we perceive. The first scale
shows this difference more clearly.

48. Since logarithms are exponents, the properties of logarithms are similar to the properties
of exponents. The Product Property states that to multiply two powers that have the same
base, add the exponents. Similarly, the logarithm of a product is the sum of the logarithms
of its factors. The Quotient Property states that to divide two powers that have the same
base, subtract their exponents. Similarly, the logarithm of a quotient is the difference of
Additional Answers for Chapter 10

the logarithms of the numerator and the denominator. The Power Property states that to
find the power of a power, multiply the exponents. Similarly, the logarithm of a power is
the product of the logarithm and the exponent. Answers should include the following.

 382   22 
5
• Quotient Property: log2 
log2 3 Replace 32 with 25 and 8 with 23.

log2 2(5  3) Quotient of Powers

5  3 or 2 Inverse Property of Exponents and Logarithms

log2 32  log2 8
log2 25  log2 23 Replace 32 with 25 and 8 with 23.

5  3 or 2 Inverse Property of Exponents and Logarithms

 382 
So, log2 
log2 32  log2 8.

Power Property: log3 94

log3 (32)4 Replace 9 with 32.

log3 3(2  4) Power of a Power

2  4 or 8 Inverse Property of Exponents and Logarithms

4 log3 9
(log3 9)  4 Commutative Property ( )

(log3 32)  4 Replace 9 with 32.

2  4 or 8 Inverse Property of Exponents and Logarithms

So, log3 94
4 log3 9.
• The Product of Powers Property and Product Property of Logarithms both involve the
addition of exponents, since logarithms are exponents.

Page 558, Lesson 10-5

log x ln x
62.    Original statement
log y ln y
log x

log x log e
   Change of Base Formula
log y log y

log e
log x log x log e log x log y
     Multiply  by the reciprocal of .
log y log e log y log e log e
log x log x

 Simplify.
log y log y

573C Chapter 10 Additional Answers

Notes

Additional Answers for Chapter 10

Chapter 10 Additional Answers 573D