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Logarithms: Definitions, Properties, and Solving Equations

Logarithms are exponents that indicate the power to which a base number must be raised to produce a given number. The document defines common, natural, and logarithms with other bases. It presents properties of logarithms including logarithms of products, quotients, exponents, and changing bases. Examples are provided of solving logarithmic equations by taking logarithms of both sides and using exponent properties to isolate the variable. Practice problems demonstrate writing logarithms in exponential form and vice versa, expanding logarithmic expressions, and solving equations.

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122 views7 pages

Logarithms: Definitions, Properties, and Solving Equations

Logarithms are exponents that indicate the power to which a base number must be raised to produce a given number. The document defines common, natural, and logarithms with other bases. It presents properties of logarithms including logarithms of products, quotients, exponents, and changing bases. Examples are provided of solving logarithmic equations by taking logarithms of both sides and using exponent properties to isolate the variable. Practice problems demonstrate writing logarithms in exponential form and vice versa, expanding logarithmic expressions, and solving equations.

Uploaded by

aaone
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Logarithms

I.

Logarithms
a. Definition: A logarithm is the exponent required to produce a
given number.
b. The General Form is: y log a x . It can be rewritten as a y x .
1.
Ex) log 2 8 3
In exponential form: 23 8
c. log a x is only defined if a and x are both positive, and a 1 .
d. The Common Logarithm has a base of 10. In general usage, the
subscript 10 is omitted when dealing with common logs.
1.
Ex) log10 1
In exponential form: 101 10
e. A Natural Logarithm is a logarithm where the base is the number
e (where e 2.72828 ). log e x is replaced with ln x . Natural
logarithms play an important role in theoretical mathematics and
the natural sciences.

II.

Properties of Logs:
Identity

Example

Log of a Product:

log 2 24 log 2 6 log 2 4

log xy log x log y

Log of a Quotient:

log

x
log x log y
y

log 2

10
log 2 10 log 2 3
3

Log with Exponent:

log 2 83 3 log 2 8
1
1
log 4 32 log32 4 log 32
4

log x a a log x
1
1
log a x log x a log x
a
Identities:

log a a 1
log a 1 0
ln e 1
ln 1 0

log 3 3 1

log 2 1 0

Negative Exponents:

log a

1
log a x
x

log 3

1
log 3 9
9

Change in base (from base a to base 10):

log a x

log 2 5

log x ln x

log a ln a

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log 5 ln 5

log 2 ln 2

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III.

Solving Equations Using Logarithms


1. 4 x 7
Take the log of both sides. You can use the
ln 4 x ln 7
common logarithm or natural log, but in
practice the natural log (or ln) is used more.
Use the exponent property to rewrite, then
divide to solve for x .

x ln 4 ln 7

ln 7
or log 4 7
ln 4


ln610 ln 11

2. 6 10 x 11x
x

Take the log of both sides.

ln 6 ln 10 x ln 11x
ln 6 x ln 10 x ln 11

Use exponent property.

ln 6 x ln 11 x ln10

Get x terms on one side.

ln 6 xln 11 ln 10

Factor out the x terms.

ln 6
ln 11 ln 10

Divide and solve for x .

3. 4e x 7
ex

7
4

Divide by 4 to get e x by itself.

ln e x ln

7
4

Take the log of both sides.

x ln e ln

7
4

Use exponent rule.

x(1) ln
x ln

7
4

Use identity that ln e 1 .

7
or x ln 7 ln 4
4

Simplify.

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Practice Problems:
Write in exponential form:
1. log 3 27 =3
2. log 2 32 =5
3. log17 1 =0

Write it logarithmic form:


4. 6 2 36
5. 103 1000

6. 9

Solve for x:
7. log x 49 2

8. log 5 125 x

9. log 8 x 2

10. e x 11

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11.

1 x
e 20
4

12. 5 x 26

13. log 16 x

1
2

14. log 7 x 1

15. log 3

1
x
9

Use the properties of logs to write in expanded form:


16. log 8 xz

u3
17. log 7 4
v

xy 2
18. log 5 4
z

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19. ln x 2 yz

20. log 7

x3
y

Solve the following equations using logarithms:


21. 5 x 6

22. e x 3

23. 2 x1 6

24. 32 x1 4

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Answers to Logarithms:
1.

33 27

13.

x4

2.

25 32

14.

3.

17 0 1

15.

x 2

4.

log 6 36 2

16.

log 8 x log 8 z

5.

log1000 3

17.

3 log 7 u 4 log 7 v

6.

log 9 3

18.

log 5 x 2 log 5 y 4 log 5 z

7.

x7

19.

2 ln x ln y ln z

8.

x3

20.

3
1
log 7 x log 7 y
2
2

9.

x 64

21.

10.

x ln 11

22.

x ln 3

11.

x ln 80

23.

12.

ln 26
ln 5

24.

1 ln 4
x
1
2 ln 3

1
2

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1
7

ln 6
or log 5 6
ln 5

ln 6
1
ln 2

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