1. Evaluate A. B.
C. 1
D. 2
5–Minute Check 1

2. Evaluate A. B.
C. 1
D. 2
5–Minute Check 1

3. Apply properties of logarithms. Apply the Change of Base Formula.
You evaluated logarithmic expressions with different bases. (Lesson 3–2)
Apply properties of logarithms.
Apply the Change of Base Formula.
Then/Now

4. Key Concept 1

5. A. Express log 96 in terms of log 2 and log 3.
Use the Properties of Logarithms
A. Express log 96 in terms of log 2 and log 3.
log 96 = log (25 ● 3) 96 = 25 ● 3
= log 25 + log 3 Product Property
= 5 log 2 + log 3 Power Property
Answer: 5 log 2 + log 3
Example 1

6. B. Express in terms of log 2 and log 3.
Use the Properties of Logarithms
B. Express in terms of log 2 and log 3.
= log 32 – log 9 Quotient Property
= log 25 – log32 25 = 32 and 32 = 9
= 5 log 2 – 2 log 3 Power Property
Answer: 5 log 2 – 2 log 3
Example 1

7. Express ln in terms of ln 3 and ln 5.
A. 3 ln ln 3
B. ln 53 – ln 33
C. 3 ln 5 – 3 ln 3
D. 3 ln 3 – 3 ln 5
Example 1

8. Express ln in terms of ln 3 and ln 5.
A. 3 ln ln 3
B. ln 53 – ln 33
C. 3 ln 5 – 3 ln 3
D. 3 ln 3 – 3 ln 5
Example 1

9. Rewrite using rational exponents.
Simplify Logarithms
A. Evaluate
Rewrite using rational exponents.
25 = 32
Power Property of Exponents
Power Property of Logarithms
logx x = 1
Answer:
Example 2

10. 3 ln e4 – 2 ln e2 = 4(3 ln e) – 2(2 ln e) Power Property of Logarithms
Simplify Logarithms
B. Evaluate 3 ln e4 – 2 ln e2.
3 ln e4 – 2 ln e2 = 4(3 ln e) – 2(2 ln e) Power Property of Logarithms
= 12 ln e – 4 ln e Multiply.
= 12(1) – 4(1) or 8 ln e = 1
Answer: 8
Example 2

11. Evaluate
A. 4 B. C. D.
Example 2

12. Evaluate
A. 4 B. C. D.
Example 2

13. The expression is the logarithm of the product of 4, m3, and n5.
Expand Logarithmic Expressions
A. Expand ln 4m3n5.
The expression is the logarithm of the product of 4, m3, and n5.
ln 4m3n5 = ln 4 + ln m3 + ln n5 Product Property
= ln ln m + 5 ln n Power Property
Answer: ln ln m + 5 ln n
Example 3

14. The expression is the logarithm of the quotient of 2x – 3 and
Expand Logarithmic Expressions
B. Expand
The expression is the logarithm of the quotient of 2x – 3 and
Quotient Property
Product Property
Rewrite using rational exponents.
Power Property
Example 3

15. Expand Logarithmic Expressions
Answer:
Example 3

16. Expand . A. 3 ln x – ln (x – 7) B. 3 ln x + ln (x – 7)
C. ln (x – 7) – 3 ln x
D. ln x3 – ln (x – 7)
Example 3

17. Expand . A. 3 ln x – ln (x – 7) B. 3 ln x + ln (x – 7)
C. ln (x – 7) – 3 ln x
D. ln x3 – ln (x – 7)
Example 3

18. A. Condense . Power Property Quotient Property Answer:
Condense Logarithmic Expressions
A. Condense
Power Property
Quotient Property
Answer:
Example 4

19. 5 ln (x + 1) + 6 ln x = ln (x + 1)5 + ln x6 Power Property
Condense Logarithmic Expressions
B. Condense 5 ln (x + 1) + 6 ln x.
5 ln (x + 1) + 6 ln x = ln (x + 1)5 + ln x6 Power Property
= ln x6(x + 1)5 Product Property
Answer: ln x6(x + 1)5
Example 4

20. Condense – ln x2 + ln (x + 3) + ln x.
A. In x(x + 3) B. C. D.
Example 4

21. Condense – ln x2 + ln (x + 3) + ln x.
A. In x(x + 3) B. C. D.
Example 4

22. Key Concept 2

23. log 6 4 = Change of Base Formula
Use the Change of Base Formula
A. Evaluate log 6 4.
log 6 4 = Change of Base Formula
≈ 0.77 Use a calculator.
Answer: 0.77
Example 5

24. = Change of Base Formula
Use the Change of Base Formula
B. Evaluate
= Change of Base Formula
≈ –1.89 Use a calculator.
Answer: –1.89
Example 5

25. Evaluate
A. –2
B. –0.5
C. 0.5
D. 2
Example 5

26. Evaluate
A. –2
B. –0.5
C. 0.5
D. 2
Example 5

27. Use the Change of Base Formula
ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function , where N1 and N2 are the numbers of each type of species found in the sample S = ( N1 + N2 ). Find the measure of diversity for environments that find 25 and 50 species in the samples.
Example 6

28. Use the Change of Base Formula
Let N1 = 25, N2 = 50, and S = 75. Substitute for the values of N1, N2, and S and solve.
D Original equation
N1 = 25, N2 = 50, and S = 75
Change of Base Formula
Example 6

29. ≈ 0.918 Use a calculator. Answer: 0.918 Use the Change of Base Formula
Example 6

30. Use the Change of Base Formula
B. ECOLOGY Diversity in a certain ecological environment containing two different species is modeled by the function , where N1 and N2 are the numbers of each type of species found in the sample S = ( N1 + N2 ). Find the measure of diversity for environments that find 10 and 60 species in the samples.
Example 6

31. Use the Change of Base Formula
Let N1 = 10, N2 = 60, and S = 70. Substitute for the values of N1, N2, and S and solve.
D Original equation
N1 = 10, N2 = 60, and S = 70
Change of Base Formula
Example 6

32. ≈ 0.592 Use a calculator. Answer: 0.592 Use the Change of Base Formula
Example 6

33. PHOTOGRAPHY In photography, exposure is the amount of light allowed to strike the film. Exposure can be adjusted by the number of stops used to take a photograph. The change in the number of stops n needed is related to the change in exposure c by n = log2c. How many stops would a photographer use to get exposure?
A. –2 stops
B. 2 stops
C. –0.5
D. 0.5
Example 6

34. PHOTOGRAPHY In photography, exposure is the amount of light allowed to strike the film. Exposure can be adjusted by the number of stops used to take a photograph. The change in the number of stops n needed is related to the change in exposure c by n = log2c. How many stops would a photographer use to get exposure?
A. –2 stops
B. 2 stops
C. –0.5
D. 0.5
Example 6