1. College Algebra Chapter 4 Exponential and Logarithmic Functions
Section 4.4
Properties of Logarithms
Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

2. Concepts
Apply the Product, Quotient, and Power Properties of Logarithms
Write a Logarithmic Expression in Expanded Form
Write a Logarithmic Expression as a Single Logarithm
Apply the Change-of-Base Formula

3. Concept 1 Apply the Product, Quotient, and Power Properties of Logarithms

4. Apply the Product, Quotient, and Power Properties of Logarithms
Let b, x, and y be positive real numbers where b ≠ 1. Product Property :
Quotient Property :
Power Property :
For these exercises, assume that all variable expressions represent positive real numbers.

5. Examples 1 – 3
Use the product property of logarithms to write the logarithm as a sum. Then simplify if possible.
log (2xy) = log 2 + log x + log y
ln (3(a + b)) = ln 3 + ln (a + b)

6. Skill Practice 1
Write the logarithm as a sum and simplify if possible. Assume that a, c, and d represented positive real numbers.

7. Examples 4 – 6
Use the quotient property of logarithms to write the logarithm as a difference. Then simplify if possible.

8. Skill Practice 2
Write the logarithm as the difference of logarithms and simplify if possible. Assume that t represents a positive real number.

9. Examples 7 – 9
Apply the power property of logarithms.

10. Skill Practice 3
Apply the power property of logarithms.

11. Concept 2 Write a Logarithmic Expression in Expanded Form

12. Example 10
Write the expression as the sum or difference of logarithms.
Solution:

13. Example 11
Write the expression as the sum or difference of logarithms.
Solution:

14. Example 12
Write the expression as the sum or difference of logarithms.
Solution:

15. Example 13
Write the expression as the sum or difference of logarithms.
Solution:

16. Example 14
Write the expression as the sum or difference of logarithms.
Solution:

17. Example 15
Write the expression as the sum or difference of logarithms.
Solution:

18. Skill Practice 4
Write the expression as the sum or difference of logarithms.

19. Concept 3 Write a Logarithmic Expression as a Single Logarithm

20. Example 16
Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. log (4x – 3) – log x Solution:

21. Example 17
Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible.
Solution:

22. Example 18
Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. 3 ln x + ln (x - 2) – ln 5 Solution:

23. Example 19
Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible.
Solution:

24. Example 20
Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible.
Solution:

25. Skill Practice 5
Write the expression as a single logarithm and simplify the result if possible.

26. Skill Practice 6
Write the expression as a single logarithms and simplify the results if possible.

27. Examples 21 – 23

28. Skill Practice 7

29. Concept 4 Apply the Change-of-Base Formula

30. Apply the Change-of-Base Formula
Let a and b be positive real numbers such that a ≠ 1 and b ≠ 1. Then for any positive real number x
In particular,

31. Examples 24, 25
Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places.

32. Skill Practice 8 Estimate between two consecutive integers.
Use the change-of-base formula to evaluate
by using base 10. Round to 4 decimal places.
Use the change-of-base formula
by using base e. Round to 4 decimal places.
Check the result by using the related exponential form.