Download 7.5 Properties of Logarithms

NAME _____________________________________________ DATE ____________________________ PERIOD _____________
7-5 Study Guide and Intervention
Properties of Logarithms
Properties of Logarithms Properties of exponents can be used to develop the following properties of logarithms.
Product Property
of Logarithms
For all positive numbers a, b, and x, where x ≠ 1,
log 𝑥 ab = log 𝑥 a + log 𝑥 b.
Quotient Property
of Logarithms
For all positive numbers a, b, and x, where x ≠ 1,
𝑎
log 𝑥 = log 𝑥 a – log 𝑥 b.
Power Property
of Logarithms
For any real number p and positive numbers m and b, where b ≠ 1,
log 𝑏 𝑚𝑝 = p log 𝑏 m.
𝑏
Example: 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 (32 · 4)
= log 3 32 + log 3 4
= 2 + log 3 4
≈ 2 + 1.2619
≈ 3.2619
b. 𝐥𝐨𝐠 𝟑 7
28
log 3 7 = log 3 ( )
4
= log 3 28 – log 3 4
≈ 3.0331 – 1.2619
≈ 1.7712
c. 𝐥𝐨𝐠 𝟑 256
log 3 256 = log 3 (44 )
= 4 · log 3 4
≈ 4(1.2619)
≈ 5.0476
Exercises
Use 𝐥𝐨𝐠 𝟏𝟐 3 ≈ 0.4421 and 𝐥𝐨𝐠 𝟏𝟐 7 ≈ 0.7831 to approximate the value of each expression.
7
1. log12 21
2. log12
4. log12 36
5. log12 63
6. log12
8. log12 16,807
9. log12 441
7. log12
81
49
3. log12 49
3
27
49
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
13. log 5 144
14. log 5 16
̅
16. log 5 1.3
17. log 5 16
Chapter 7
12. log 5 0.75
27
15. log 5 375
9
18. log 5
33
81
5
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
7-5 Study Guide and Intervention (continued)
Properties of Logarithms
Solve Logarithmic Equations You can use the properties of logarithms to solve equations involving logarithms.
Example: Solve each equation.
a. 2 𝐥𝐨𝐠 𝟑 x – 𝐥𝐨𝐠 𝟑 4 = 𝐥𝐨𝐠 𝟑 25
2 log 3 x – log 3 4 = log 3 25
2
log 3 𝑥 – log 3 4 = log 3 25
log 3
𝑥2
4
𝑥2
4
2
Original equation
Power Property
= log 3 25
Quotient Property
= 25
Property of Equality for Logarithmic Functions
𝑥 = 100
x = ±10
Multiply each side by 4.
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
3
x(x + 2) = 2
Definition of logarithm
𝑥 2 + 2x = 8
Distributive Property
2
𝑥 + 2x – 8 = 0
(x + 4)(x – 2) = 0
x = 2 or x = –4
Subtract 8 from each side.
Factor.
Zero Product Property
Since logarithms are undefined for x < 0, –4 is an extraneous solution. The only solution is 2.
Exercises
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
1
3. 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 (4c – 1) = log 3 5
10. log 5 (x + 3) – log 5 (2x – 1) = 2
Chapter 7
34
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
7-5 Skills Practice
Properties of Logarithms
Use 𝐥𝐨𝐠 𝟐 3 ≈ 1.5850 and 𝐥𝐨𝐠 𝟐 5 ≈ 2.3219 to approximate the value of each expression.
1. log 2 25
2. log 2 27
3
5
3. log 2 5
4. log 2
5. log 2 15
6. log 2 45
7. log 2 75
8. log 2 0.6
1
3
9
9. log 2 3
10. log 2 5
Solve each equation. Check your solutions.
11. log10 27 = 3 log10 x
12. 3 log 7 4 = 2 log 7 b
13. log 4 5 + log 4 x = log 4 60
14. log 6 2c + log 6 8 = log 6 80
15. log 5 y – log 5 8 = log 5 1
16. log 2 q – log 2 3 = log 2 7
17. log 9 4 + 2 log 9 5 = log 9 w
18. 3 log 8 2 – log 8 4 = log 8 b
19. log10 x + log10 (3x – 5) = log10 2
20. log 4 x + log 4 (2x – 3) = log 4 2
21. log 3 d + log 3 3 = 3
22. log 10 y – log10 (2 – y) = 0
23. log 2 r + 2 log 2 5 = 0
24. log 2 (x + 4) – log 2 (x – 3) = 3
25. log 4 (n + 1) – log 4 (n – 2) = 1
26. log 5 10 + log 5 12 = 3 log 5 2 + log 5 a
Chapter 7
34
Glencoe Algebra 2