1. Section 5.4 Properties of Logarithmic Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

2. Objectives  Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.  Simplify expressions of the type log a a x and

3. Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a, log a MN = log a M + log a N. (The logarithm of a product is the sum of the logarithms of the factors.)

4. Example Express as a single logarithm:

5. Logarithms of Powers The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.)

6. Example Express as a product.

7. Logarithms of Quotients The Quotient Rule For any positive numbers M and N, and any logarithmic base a, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)

8. Example Express as a difference of logarithms:

9. Example Express as a single logarithm:

10. Example Express each of the following in terms of sums and differences of logarithms.

11. Example (continued)

13. Example Express as a single logarithm:

14. Example Given that log a 2 ≈ 0.301 and log a 3 ≈ 0.477, find each of the following, if possible.

15. Examples (continued) Cannot be found using these properties and the given information.

16. Expressions of the Type log a a x The Logarithm of a Base to a Power For any base a and any real number x, log a a x = x. (The logarithm, base a, of a to a power is the power.)

17. Examples Simplify. a) log a a 8 b) ln e  t c) log 10 3k a. log a a 8

18. Expressions of the Type A Base to a Logarithmic Power For any base a and any positive real number x, (The number a raised to the power log a x is x.)

19. Example Simplify.