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# Logarithms Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Cancellation Properties of Logarithms

The first two logarithm properties we will learn are the cancellation properties. They come directly from rewriting one form of an equation in the other form.

log a a x = x and a log a x = x

When the bases of the exponent and logarithm are the same, they cancel. Let us see why these properties are true. What would the expression log a a x be? We will rewrite the equation “log a a x = ?” as an exponential equation: a ? = a x . Now we can see that “?” is x . This is why log a a x = x . What would a log a x be? We will rewrite “ a log a x = ?” as a logarithmic equation: log a ? = log a x , so “?” is x , and a log a x = x .

#### Examples

• 5 log 5 2
• The bases of the logarithm and exponent are both 5, so 5 log 5 2 simplifies to 2.

10 log 10 8 = 8       4 log 4 x = x e log e 6 = 6

29 log 29 1 = 1   log m m r = r    log 7 7 ab = ab

Sometimes we need to use exponent properties before using the property log a a x = x .

## Natural and Common Logs

Two types of logarithms occur frequently enough to have their own notation. They are log e and log 10 . The notation for log e is “ln” (pronounced “ell-in”) and is called the natural log . The notation for log 10 is “log” (no base is written) and is called the common log . The cancel properties for these special logarithms are

ln e x = x e ln x = x        and       log 10 x = x      10 log x = x .

#### Examples

• e 4 = x − 1 rewritten as a log equation is ln( x − 1) = 4
• 10 x = 6 rewritten as a log equation is log 6 = x
• ln 2 x = 25 rewritten as an exponent equation is e 25 = 2 x
• log(2 x − 9) = 4 rewritten as an exponent equation is 10 4 = 2 x − 9
• ln e 15 = 15
• e ln 14 = 14
• ln e −4 = −4
• 10 log 5 = 5
• log 10 −4 = −4

## Logarithm Practice Problem

#### Practice

1. Rewrite as a logarithm: e 3 x = 4
2. Rewrite as a logarithm: 10 x −1 = 15
3. Rewrite as an exponent: ln 6 = x + 1
4. Rewrite as an exponent: log 5 x = 3

Use logarithm properties to simplify the expression.

1. 9 log 9 3
2. 10 log 10 14
3. 5 log 5 x
4. log 15 15 2
5. log 10 10 −8
6. log e e x
7. ln e 5
8. 10 log 9
9. e ln 6
10. log 10 3 x −1
11. ln e x +1

#### Solutions

1. ln 4 = 3x

2. log 15 = x − 1

3. e x +1 = 6

4. 10 3 = 5 x

5. 9 log 9 3 = 3

6. 10 log 10 14 = 14

7. 5 log 5 x = x

8. log 15 15 2 = 2

9. log 10 10 −8 = −8

10. log e e x = x

11.

12.

13.

14.

15.

16.

17. ln e 5 = 5

18. 10 log 9 = 9

19. e ln 6 = 6

20. log 10 3 x −1 = 3 x − 1

21. ln e x +1 = x + 1

Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.

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