INSTRUCTIONAL PLANNING
(The process of systematically planning, developing, evaluating, and managing the instructional process by using
principles of teaching and learning – D.O. 42 s. 2016)
Detailed Lesson Plan (DLP)
DLP No. 31 Learning Area: General Mathematics Grade Level : 11 Quarter: I Duration: 1 hr
Learning
The learner illustrates the laws of logarithms. Code:M11GM-Ih-3
Competency/ies
Key Concepts/
Laws of Logarithm and its proof; expand, condense or simplifylogarithmic expressions using the
Understanding to
laws of logarithms and change-base-formula
be Developed:
Knowledge: Illustrate the laws of logarithms and its proof and the change-base-formula.
1. Learning Skills: Apply the laws of logarithms or change-based-formula to expand, condense or
Objectives: simplifylogarithmic expressions.
Attitudes: Show obedience in applying the laws of logarithms and change-base-formula.
Values: Obedience, cooperation
2. Content LAWS OF LOGARITHMS
3 .Learning Learners Material, Teaching Guide, Worksheets, Meta strips, Hand-outs, Quiz
Resources
4. Procedure
*Drill. Recap the previous lesson.
4.1 Introductory Activity (7
Singing the song “My TatlongBebe” and pass a “Cabbage ball” containing the
min.)
questions of the previous lesson “Evaluating logarithms”.
* Activity by Group
Find the value of the following logarithmic expressions. (each logarithm is written
in meta strips)
1. log 7 (73 ∙ 78 ) 6. log 7 73 + log 7 78
49
2. log 7 ( 7 ) 7. log 7 49 − log 7 7
3. log 7 75 8. 5 log 7 7
4. log 3 (27 ∙ 81) 9. log 3 27 + log 3 81
4.2 Activity (10 min.) 24
5. log 2 ( ) 10. log 2 24 − log 2 210
210
Whichpairs of logarithms have the same answers? Pair the logarithms having the
same number?
What can you say about these logarithms having the same answer?
*Mention that these pairs of logarithms can be generalized into certain Laws of
Logarithms.
*Introduce the Laws of Logarithms based from the previous activity or let the student
complete the law.
LAWS OF LOGARITHMS
Let 𝑏 > 0, 𝑏 ≠ 1 and let 𝑛 ∈ 𝑅. For 𝑢 > 0 𝑎𝑛𝑑 𝑣 > 0, then
1. log 𝑏 (𝑢𝑣) = log 𝑏 𝑢 + log 𝑏 𝑣
4.3. Analysis (20 min.) 𝑢
2. log 𝑏 (𝑣 ) = log 𝑏 𝑢 − log 𝑏 𝑣
3. log 𝑏 𝑢𝑛 = 𝑛 log 𝑏 𝑢
*To illustrate, ask students to choose pairs of logarithms as examples to these
properties/laws.
CHANGE OF BASE FORMULA
Let 𝑎, 𝑏, and 𝑥 be positive real numbers, with 𝑎 ≠ 1 𝑎𝑛𝑑 𝑏 ≠ 1, then
log 𝑎 𝑥
log 𝑏 𝑥 =
log 𝑎 𝑏
Discusss the proof of the laws of logarithms and illustrates by giving examples.
(Refer to the hand outs)
*Activity by pair (Think-pair-Share) - Worksheets
A. Use the laws of logarithms to expand the expression as a sum, difference, or
multiples of logarithms. Simplify.
3
1. log(𝑎𝑏 2 )2. ln[𝑥(𝑥 − 5)]3. log 3 (𝑥)3
B. Use the laws of logarithms to condense into a single logarithm.
1. log 2 + log 3 3. log 5 𝑥 2 − 3 log 𝑥
2. 2 𝑙𝑛 𝑥 − log 3 4. 2 − log 5
C. Use the change-of-base formula to rewrite the following logarithmic expressions to
the indicated base, and then compute the value.
1. log 8 32 (change to base 2)
1
2. log 243 27 (change to base 3)
How did you answer the activity?
What are the properties or laws did you apply?
1. What are the laws of the logarithms?Illustrates by giving an example.
8𝑥 2
4.4 Abstraction (5 min.) 2. What law/s applied in expanding the logarithmic expressionlog 2 ( )?
𝑦
*Group Activity. Assign one item/equation to each group to present and discuss their
answer to the class.
True or False. Apply the laws of logarithms to determine whether the given
logarithmic equation is true or false.
1. (log 3 2)(log 3 4) = log 3 8
2. log 22 = (log 2)2
log 𝑥
3. log 4 (𝑥 − 4) = 4
log4 4
4.5 Application (5 min.) 4. log 3 2𝑥 2 = log 3 2 + log 3 𝑥 2
5. 3 (log 9 𝑥 2 ) = 6 log 9 𝑥
Problem 1.
The magnitude R of an earthquake using a Richter Scale is given by
2 𝐸
𝑅 = 3 log 104.40 , where E (in joules) is the energy released by the earthquake.
Suppose that the earthquake hit in Bohol in 2013 released 1015.2 joules of energy,
Using the richter scale, what is the magnitude of the earthquake hit in Bohol. (Use
the laws of logarithms to solve the problem.
Quiz.
4.6 Assessment (10 min.) A. Use the laws of logarithms to expand the expression as a sum, difference, or
multiples of logarithms. Simplify.
𝑥2 4𝑥 2
1. log 2. log 4 (16𝑎)3. log 2 ( )
2 𝑦
B. Use the laws of logarithms to condense into a single logarithm.
1. log(𝑥 + 2) + log(𝑥 − 2)
2. 2 log 3 5 + 1
3. 2 ln(3) − ln 4
C. True or False.
1. log 5 25𝑥 2 = 2 + 2 log 5 𝑥
D. Change log 6 4 to base 2 and simplify.
Given log 5 2 ≈ 0.431. Use this fact and the laws of logarithms to approximate the
values of:
4.7 Assignment (2 min.) 1
(a) log 5 8 (b) log 5 (c) log 5 √2
16
“As the world continually multiplies, are we in a generation where are divided, or
4.8 Concluding Activity (1 people are equal.”
min.) By Anthony Liccione
5. Remarks
6. Reflections
A. No. of learners who earned 80% in the
evaluation.
B. No. of learners who require additional
activities for remediation.
C. Did the remedial lessons work? No. of
learners who have caught up with the
lesson.
D. No. of learners who continue to
require remediation.
E. Which of my learning strategies
worked well? Why did these work?
F. What difficulties did I encounter which
my principal or supervisor can help me
solve?
G. What innovation or localized
materials did I use/discover which I
wish to share with other teachers?
Prepared by:
Name: CHARLYN E. CARPENTERO-SATIRA School: MAXIMINO NOEL MEMORIAL NATIONAL HIGH SCHOOL
Position/Designation: TEACHER II Division: CARCAR CITY DIVISION
Contact Number: 09750160664 Email address: enchailygne@gmail.com
Bibliography
Attachments:
1. Questions and Answers in the Cabbage Ball
2. Questions and Answers in the Metastrips
3. Handouts
4. Worksheets
5. Quiz
Attachment 1.
Question and Answer in the Cabbage Ball
(a) log 2 32 Ans. 5
(b) log 9 729 Ans. 3
(c) log 5 5 Ans. 1
(d) log 1 16 Ans. -4
2
(e) log 7 1 Ans.0
1
(f) log 5 Ans. -1/2
√5
Attachment 2.
Problems in metastrips.
Find the value of the following logarithmic expressions and pair the logarithms with the same answer.
(each logarithm is written in meta strips)
1. log 7(73 ∙ 78 ) 6. log 7 73 + log 7 78 Ans. (1) and (6) is 11
49
2. log 7 ( ) 7. log 7 49 − log 7 7 Ans. (2) and (7) is 1
7
3. log 7 75 8. 5 log 7 7 Ans. (3) and (8) is 5
4. log 3 (27 ∙ 81) 9. log 3 27 + log 3 81 Ans. (4) and (9) is -6
24
5. log 2 (210 )10. log 2 24 − log 2 210 Ans. (5) and (10) is 7
Attachment 3. Handouts
LAWS OF LOGARITHMS
Let 𝑏 > 0, 𝑏 ≠ 1 and let 𝑛 ∈ 𝑅. For 𝑢 > 0 𝑎𝑛𝑑 𝑣 > 0, then
4. log 𝑏 (𝑢𝑣) = log 𝑏 𝑢 + log 𝑏 𝑣
𝑢
5. log 𝑏 ( ) = log 𝑏 𝑢 − log 𝑏 𝑣
𝑣
𝑛
6. log 𝑏 𝑢 = 𝑛 log 𝑏 𝑢
Examples.
Proof:
Examples.
CHANGE OF BASE FORMULA
Let 𝑎, 𝑏, and 𝑥 be positive real numbers, with 𝑎 ≠ 1 𝑎𝑛𝑑 𝑏 ≠ 1, then
log 𝑥
log 𝑏 𝑥 = log𝑎 𝑏
𝑎
Examples.
Use the change-of-base formula to rewrite the following logarithmic expressions to the indicated base,
and then compute the value.
log2 32 5
1. log 8 32 (change to base 2) Solution: log 8 32 = log2 8
=3
1
1 1 log3 −3
27
2. log 243 27 (change to base 3) Solution: log 8 27 = log =
3 243 5
Attachment 4. (Worksheets)
True or False.Apply the laws of logarithms to determine whether the given logarithmic equation is true
or false.
1. (log 3 2)(log 3 4) = log 3 8 Ans. False
2. log 22 = (log 2)2 Ans. False
log 𝑥
3. log 4 (𝑥 − 4) = log4 4 Ans. True
4
4. log 3 2𝑥 2 = log 3 2 + 2 log 3 𝑥 Ans. True
5. 3 (log 9 𝑥 2 ) = 6 log 9 𝑥 Ans. True
6. 3 (log 9 𝑥 2 ) = 6 log 9 𝑥
Problem 1.
The magnitude R of an earthquake using a Richter Scale is given by
2 𝐸
𝑅 = 3 log 104.40 , where E (in joules) is the energy released by the earthquake.
Suppose that the earthquake hit in Bohol in 2013 released 1015.2 joules of energy, using the richter scale,
what is the magnitude of the earthquake hit in Bohol. (Use the law of logarithms to solve the problem.
Solution.
2 1015.2 2 2
𝑅 = 3 log = (log 1015.2
104.4 3
− log 104.4 ) = 3 (15.2 − 4.4) = 7.2
Attachment 5. Quiz
A. Use the laws of logarithms to expand the expression as a sum, difference, or multiples of logarithms. Simplify.
𝑥2
1. log Ans. 2 log 𝑥 − log 2
2
2. log 4 (16𝑎) Ans. 2 + log 4 𝑎
4𝑥 2
3. log 2 ( ) Ans. 2 + 2 log 2 𝑥 − log 2 𝑦
𝑦
B. Use the laws of logarithms to condense into a single logarithm.
𝑥+2
1. log(𝑥 + 2) + log(𝑥 − 2) Ans. log ( )
𝑥−2
2. 2 log 3 5 + 1 Ans. log 3 75 =
3. 2 ln(3) − ln 4 Ans. ln 36
C. True or False.
1. log 5 25𝑥 2 = 2 + 2 log 5 𝑥 Ans. True
log 4 2
D. Change log 6 4 to base 2. Ans. log2 6 = log
2 26
