1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 1 Chapter 5 Logarithmic Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 2 5.6 More Properties of Logarithms

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 3 Product Property for Logarithms For x > 0, y > 0, b > 0, and b ≠ 1, log b (x) + log b (y) = log b (xy) In words, the sum of logarithms is the logarithm of the product of their inputs.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 4 Example: Using the Product Property for Logarithms Simplify. Write the sum of logarithms as a single logarithm. 1. log b (2x) + log b (x)2. 3 log b (x 2 ) + 2 log b (6x)

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 5 Solution 1. 2.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 6 Applying Properties Warning To apply the product property for logarithms, the coefficient of each logarithm must be 1. You may need to apply the power property first!

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 7 Quotient Property for Logarithms For x > 0, y > 0, b > 0, and b ≠ 1, In words, the difference of two logarithms is the logarithm of the quotient of their inputs.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 8 Example: Product and Quotient Properties Simplify. Write the result as a single logarithm with a coefficient of 1. 1. 1og b (6w 6 ) – log b (w 2 ) 2. 2 log b (3p) + 3 log b (p 2 ) – 4 log b (2p)

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 9 Solution 1.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 10 Solution 2.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 11 Example: Solving a Logarithmic Equation Solve

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 12 Solution

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 13 Change-of-Base Property For a > 0, b > 0, a ≠ 1, b ≠ 1, and x > 0,

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 14 Change-of-Base To find a logarithm to a base other than 10, we use the change-of-base property to convert to log 10 ; then we can use the log key on a calculator.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 15 Example: Converting to log 10 Find log 2 (12).

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 16 Solution Using the log key on a calculator, we compute So, log 2 (12) ≈ 3.5850.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 17 Example: Using the Change-of-Base Property Write as a single logarithm.

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 18 Solution

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 19 Example: Using a Graphing Calculator to Graph a Logarithmic Function Using a graphing calculator to draw the graph of y = log 3 (x).

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 20 Solution Use the change-of-base property, Using the log key on the graphing calculator, enter the function and draw the graph.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.6, Slide 21 Comparing Properties of Logarithms Warning It is common to confuse the quotient property and the change-of-base property for logarithms. In general, and