Logarithms Help

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

Examples

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

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.

5. 9 ^{log ₉ 3}
6. 10 ^{log ₁₀ 14}
7. 5 ^{log ₅ x}
8. log ₁₅ 15 ²
9. log ₁₀ 10 ⁻⁸
10. log _e e ^x
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. ln e ⁵
19. 
20. 10 ^{log 9}
21. e ^{ln 6}
22. log 10 ^{3 x −1}
23. ln e ^{x +1}

Solutions

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