2. Copyright © 2009 Pearson Education, Inc. CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest

3. Copyright © 2009 Pearson Education, Inc. 5.4 Properties of Logarithmic Functions  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.

4. Slide 5.4 - 4 Copyright © 2009 Pearson Education, Inc. 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.)

5. Slide 5.4 - 5 Copyright © 2009 Pearson Education, Inc. Example Express as a single logarithm: Solution:

6. Slide 5.4 - 6 Copyright © 2009 Pearson Education, Inc. 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.)

7. Slide 5.4 - 7 Copyright © 2009 Pearson Education, Inc. Examples Express as a product. Solution:

8. Slide 5.4 - 8 Copyright © 2009 Pearson Education, Inc. 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.)

9. Slide 5.4 - 9 Copyright © 2009 Pearson Education, Inc. Examples Express as a difference of logarithms: Solution:

10. Slide 5.4 - 10 Copyright © 2009 Pearson Education, Inc. Examples Express as a single logarithm: Solution:

11. Slide 5.4 - 11 Copyright © 2009 Pearson Education, Inc. Applying the Properties - Example Express each of the following in terms of sums and differences of logarithms. Solution:

12. Slide 5.4 - 12 Copyright © 2009 Pearson Education, Inc. Example (continued) Solution:

13. Slide 5.4 - 13 Copyright © 2009 Pearson Education, Inc. Example (continued) Solution:

14. Slide 5.4 - 14 Copyright © 2009 Pearson Education, Inc. Example Express as a single logarithm: Solution:

15. Slide 5.4 - 15 Copyright © 2009 Pearson Education, Inc. Examples Given that log a 2 ≈ 0.301 and log a 3 ≈ 0.477, find each of the following, if possible. Solution:

16. Slide 5.4 - 16 Copyright © 2009 Pearson Education, Inc. Examples (continued) Solution: Cannot be found using these properties and the given information.

17. Slide 5.4 - 17 Copyright © 2009 Pearson Education, Inc. 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.)

18. Slide 5.4 - 18 Copyright © 2009 Pearson Education, Inc. Examples Simplify. a) log a a 8 b) ln e  t c) log 10 3k Solution: a. log a a 8

19. Slide 5.4 - 19 Copyright © 2009 Pearson Education, Inc. 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.)

20. Slide 5.4 - 20 Copyright © 2009 Pearson Education, Inc. Examples Simplify. Solution: