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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Lesson 7-5

NAME DATE PERIOD

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Chapter 7 33 Glencoe Algebra 2

Properties of Logarithms Properties of exponents can be used to develop the following properties of logarithms.Product Property of LogarithmsFor all positive numbers a, b, and x, where x ≠ 1, log x ab = log x a + log x b.

Quotient Property

of LogarithmsFor all positive numbers a, b, and x, where x ≠ 1, log x a b = log x a - log x b.

Power Property

of LogarithmsFor any real number p and positive numbers m and b, where b ≠ 1, log b m p = p log b m.

Use log

3 28

3.0331 and log

3 4

1.2619 to approximate

the value of each expression. a. log 3 36
log 3

36 = log

3 (3 2

· 4)

= log 3 32
+ log 3 4 = 2 + log 3 4 ≈ 2 + 1.2619 ≈ 3.2619b. log 3 7 log 3

7 = log

3 28
4 = log 3 28
log 3 4 ≈ 3.0331 - 1.2619 ≈ 1.7712c. log 3 256
log 3

256 = log

3 (4 4 = 4 · log3 4 ≈ 4(1.2619) ≈ 5.0476

Use log

12 3 ≈ 0.4421 and log 12 7 ≈ 0.7831 to approximate the value of each expression.

1. log

12

21 2. log

12 7 3

3. log

12 49

4. log

12

36 5. log

12

63 6. log

12 27
49
7. log12 81
49

8. log

12

16,807 9. log

12 441

Use log

5 3 ≈ 0.6826 and log 5 4 ≈ 0.8614 to approximate the value of each expression.

10. log

5

12 11. log

5

100 12. log

5 0.75

13. log

5

144 14. log

5 27
16

15. log

5 375

16. log5

1.

3 17. log

5 9 16

18. log

5 81
5

Exercises

Example

Study Guide and Intervention

Properties of Logarithms

7-5

1.22520.34101.5662

1.4421

1.6673-0.2399

0.20223.9155

2.4504

1.54402.8614-0.1788

3.0880

0.32503.6826

0.1788

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NAME DATE PERIOD

PDF Pass

Chapter 7 34 Glencoe Algebra 2

Solve Logarithmic Equations You can use the properties of logarithms to solve equations involving logarithms.

Solve each equation.

a. 2 log 3 x - log 3 4 log 3 25
2 log 3 x - log 3 4 log 3

25 Original equation

log 3 x 2 - log 3 4 log 3

25 Power Property

log 3 x 2 4 = log 3

25 Quotient Property

x 2 4 = 25 Property of Equality for Logarithmic Functions x 2 = 100 Multiply each side by 4. x = ±10 Take the square root of each side.

Since logarithms are undefined for

x < 0, -10 is an extraneous solution.

The only solution is 10.

b. log 2 x + log 2 (x + 2) = 3 log 2 x + log 2 (x + 2) = 3 Original equation log 2 x(x + 2) = 3 Product Property x(x + 2) = 2 3

De? nition of logarithm

x 2 + 2x = 8 Distributive Property x 2 + 2x - 8 = 0 Subtract 8 from each side. (x + 4)(x - 2) = 0 Factor. x = 2 or x = -4 Zero Product Property

Since logarithms are undefined for

x < 0, -4 is an extraneous solution.

The only solution is 2.

Solve each equation. Check your solutions.

1. log

5 4 log 5

2x = log

5

24 2. 3 log

4 6 log 4 8 log 4 x 3. 1 2 log 6 25
log 6 x = log 6

20 4. log

2 4 log 2 (x + 3) = log 2 8

5. log

6

2x - log

6 3 log 6 (x - 1) 6. 2 log 4 (x + 1) = log 4 (11 x)

7. log

2 x - 3 log 2 5 2 log 2

10 8. 3 log

2 x - 2 log 2

5x = 2

9. log

3 (c + 3) - log 3 (4 c - 1) = log 3

5 10. log

5 (x + 3) - log 5 (2 x - 1) = 2

Example

Exercises

Study Guide and Intervention (continued)

Properties of Logarithms

7-5 327
4 5 2 3 2

12,500

100
4 7 8 19

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