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9-2
NAME ______________________________________________ DATE______________ PERIOD _____
9-2
Skills Practice
Practice Logarithms and Logarithmic Functions
Logarithms and Logarithmic Functions Write each equation in logarithmic form. 1. 23 ⫽ 8 log2 8 ⫽ 3
2. 32 ⫽ 9 log3 9 ⫽ 2
1 64
1 64
Write each equation in logarithmic form.
冢 冣
1 2 1 ⫽ᎏ 3 9
3. 8⫺2 ⫽ ᎏ log8 ᎏ ⫽ ⫺2
4. ᎏ
1. 53 ⫽ 125 log5 125 ⫽ 3
1 81
1 ᎏᎏ
ᎏ1ᎏ
1 64
⫽ᎏ
6. 7776 5 ⫽ 6
1 5
1 64
logᎏ14ᎏ ᎏ ⫽ 3
log7776 6 ⫽ ᎏ
Write each equation in exponential form.
6. log4 64 ⫽ 3 43 ⫽ 64
7. log6 216 ⫽ 3 63 ⫽ 216
1 25
10. log10 0.00001 ⫽ ⫺5
8. log5 ᎏ ⫽ ⫺2 5⫺2 ⫽ ᎏ 1 25
3
3. 34 ⫽ 81 log3 81 ⫽ 4
8. log2 64 ⫽ 6 26 ⫽ 64 1 2
11. log25 5 ⫽ ᎏ 1 ᎏᎏ 2
10⫺5 ⫽ 0.00001
1 81
9. log3 ᎏ ⫽ ⫺4 3⫺4 ⫽ ᎏ 1 81
3 5
12. log32 8 ⫽ ᎏ 3 ᎏᎏ 5
25 ⫽ 5
32 ⫽ 8
Evaluate each expression.
1 12. log125 5 ᎏ 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 64
Evaluate each expression.
1 625
13. log4 ᎏ ⫺3
14. log5 ᎏ ⫺4
15. log8 83 3
16. log27 ᎏ ⫺ ᎏ
1 3
1 3
Solve each equation or inequality. Check your solutions. 17. log3 x ⫽ 5 243
18. log2 x ⫽ 3 8
19. log4 y ⬍ 0 0 ⬍ y ⬍ 1
20. logᎏ14ᎏ x ⫽ 3 ᎏ
1 64
Glencoe Algebra 2
1 21. log2 n ⬎ ⫺2 n ⬎ ᎏ 4
1 22. logb 3 ⫽ ᎏ 2
23. log6 (4x ⫹ 12) ⫽ 2 6
24. log2 (4x ⫺ 4) ⬎ 5 x ⬎ 9
25. log3 (x ⫹ 2) ⫽ log3 (3x) 1 Chapter 9
9
26. log6 (3y ⫺ 5) ⱖ log6 (2y ⫹ 3) y ⱖ 8
16
Glencoe Algebra 2
13. log3 81 4
14. log10 0.0001 ⫺4
17. log9 1 0
18. log8 4 ᎏ
2 3
16. logᎏ13ᎏ 27 ⫺3
1 49
20. log6 64 4
19. log7 ᎏ ⫺2
1 256
1 3
21. log3 ᎏ ⫺1
1 16
15. log2 ᎏ ⫺4
22. log4 ᎏ ⫺4
23. log9 9(n ⫹ 1) n ⫹ 1 24. 2log2 32 32
Solve each equation or inequality. Check your solutions.
1 1000
3 2
25. log10 n ⫽ ⫺3 ᎏ
26. log4 x ⬎ 3 x ⬎ 64
27. log4 x ⫽ ᎏ 8
28. logᎏ15ᎏ x ⫽ ⫺3 125
29. log7 q ⬍ 0 0 ⬍ q ⬍ 1
30. log6 (2y ⫹ 8) ⱖ 2 y ⱖ 14
1 31. logy 16 ⫽ ⫺4 ᎏ 2
32. logn ᎏ ⫽ ⫺3 2
1 8
33. logb 1024 ⫽ 5 4
34. log8 (3x ⫹ 7) ⬍ log8 (7x ⫹ 4) 35. log7 (8x ⫹ 20) ⫽ log7 (x ⫹ 6) 36. log3 (x2 ⫺ 2) ⫽ log3 x
3 4
x⬎ᎏ
⫺2
2
37. SOUND An equation for loudness, in decibels, is L ⫽ 10 log10 R, where R is the relative intensity of the sound. Sounds that reach levels of 120 decibels or more are painful to humans. What is the relative intensity of 120 decibels? 1012 38. INVESTING Maria invests $1000 in a savings account that pays 4% interest compounded annually. The value of the account A at the end of five years can be determined from the equation log A ⫽ log[1000(1 ⫹ 0.04)5]. Find the value of A to the nearest dollar. $1217 Chapter 9
Answers
17
Glencoe Algebra 2
(Lesson 9-2)
A7
11. log10 1000 3
1 2
10. log9 3 ᎏ
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9. log5 25 2
Answers
1 2
7. log9 3 ⫽ ᎏ 9 2 ⫽ 3
5. ᎏ
log3 ᎏ ⫽ ⫺4
Write each equation in exponential form. 5. log3 243 ⫽ 5 35 ⫽ 243
冢 41 冣
1 81
4. 3⫺4 ⫽ ᎏ
1 9
logᎏ13ᎏ ᎏ ⫽ 2
2. 70 ⫽ 1 log7 1 ⫽ 0
Lesson 9-2
Chapter 9
NAME ______________________________________________ DATE______________ PERIOD _____
9-3
Study Guide and Intervention
NAME ______________________________________________ DATE______________ PERIOD _____
9-3
(continued)
Skills Practice
Properties of Logarithms Solve Logarithmic Equations
Properties of Logarithms Use log2 3 ⬇ 1.5850 and log2 5 ⬇ 2.3219 to approximate the value of each expression.
You can use the properties of logarithms to solve
equations involving logarithms. Example
1. log2 25 4.6438
Solve each equation.
a. 2 log3 x ⫺ log3 4 ⫽ log3 25 2 log3 x ⫺ log3 4 ⫽ log3 25 log3 x2 ⫺ log3 4 ⫽ log3 25 x2 4 x2 ᎏ ⫽ 25 4
log3 ᎏ ⫽ log3 25
5 3
3 5
Original equation
3. log2 ᎏ ⫺0.7369
4. log2 ᎏ 0.7369
5. log2 15 3.9069
6. log2 45 5.4919
7. log2 75 6.2288
8. log2 0.6 ⫺0.7369
Power Property Quotient Property Property of Equality for Logarithmic Functions
1 3
2. 3 log4 6 ⫺ log4 8 ⫽ log4 x 27
5 2
4. log2 4 ⫺ log2 (x ⫹ 3) ⫽ log2 8 ⫺ ᎏ
5. log6 2x ⫺ log6 3 ⫽ log6 (x ⫺ 1) 3
6. 2 log4 (x ⫹ 1) ⫽ log4 (11 ⫺ x) 2
7. log2 x ⫺ 3 log2 5 ⫽ 2 log2 10 12,500
8. 3 log2 x ⫺ 2 log2 5x ⫽ 2 100
8 19
9. log3 (c ⫹ 3) ⫺ log3 (4c ⫺ 1) ⫽ log3 5 ᎏ
4 7
10. log5 (x ⫹ 3) ⫺ log5 (2x ⫺ 1) ⫽ 2 ᎏ
22
12. 3 log7 4 ⫽ 2 log7 b 8
13. log4 5 ⫹ log4 x ⫽ log4 60 12
14. log6 2c ⫹ log6 8 ⫽ log6 80 5
15. log5 y ⫺ log5 8 ⫽ log5 1 8
16. log2 q ⫺ log2 3 ⫽ log2 7 21
17. log9 4 ⫹ 2 log9 5 ⫽ log9 w 100
18. 3 log8 2 ⫺ log8 4 ⫽ log8 b 2
19. log10 x ⫹ log10 (3x ⫺ 5) ⫽ log10 2 2
20. log4 x ⫹ log4 (2x ⫺ 3) ⫽ log4 2 2
21. log3 d ⫹ log3 3 ⫽ 3 9
22. log10 y ⫺ log10 (2 ⫺ y) ⫽ 0 1
1 25
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
3. ᎏ log6 25 ⫹ log6 x ⫽ log6 20 4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check your solutions.
11. log10 27 ⫽ 3 log10 x 3
23. log2 s ⫹ 2 log2 5 ⫽ 0 ᎏ
24. log2 (x ⫹ 4) ⫺ log2 (x ⫺ 3) ⫽ 3 4
25. log4 (n ⫹ 1) ⫺ log4 (n ⫺ 2) ⫽ 1 3
26. log5 10 ⫹ log5 12 ⫽ 3 log5 2 ⫹ log5 a 15
Chapter 9
23
Glencoe Algebra 2
(Lesson 9-3)
Exercises
1 2
10. log2 ᎏ 0.8481
Solve each equation. Check your solutions.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A10
Original equation log2 x ⫹ log2 (x ⫹ 2) ⫽ 3 log2 x(x ⫹ 2) ⫽ 3 Product Property 3 x(x ⫹ 2) ⫽ 2 Definition of logarithm 2 x ⫹ 2x ⫽ 8 Distributive Property 2 x ⫹ 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.
1. log5 4 ⫹ log5 2x ⫽ log5 24 3
9 5
9. log2 ᎏ ⫺1.5850
b. log2 x ⫹ log2 (x ⫹ 2) ⫽ 3
Answers
x2 ⫽ 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.
Chapter 9
2. log2 27 4.755
Lesson 9-3
Chapter 9
NAME ______________________________________________ DATE______________ PERIOD _____
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9-3
NAME ______________________________________________ DATE______________ PERIOD _____
9-3
Practice
Word Problem Practice
Properties of Logarithms
Properties of Logarithms 1. MENTAL COMPUTATION Jessica has memorized log5 2 ⬇ 0.4307 and log5 3 ⬇ 0.6826. Using this information, to the nearest thousandth, what power of 5 is equal to 6?
Use log10 5 ⬇ 0.6990 and log10 7 ⬇ 0.8451 to approximate the value of each expression. 7 5
5 7
4. log10 ᎏ ⫺0.1461
1. log10 35 1.5441
2. log10 25 1.3980
3. log10 ᎏ 0.1461
5. log10 245 2.3892
6. log10 175 2.2431
7. log10 0.2 ⫺0.6990 8. log10 ᎏ 0.5529
3 2
10. log10 u ⫽ ᎏ log10 4 8
13. log9 (3u ⫹ 14) ⫺ log9 5 ⫽ log9 2u 2
14. 4 log2 x ⫹ log2 5 ⫽ log2 405 3
where C is the concentration of hydrogen ions. If the concentration of hydrogen ions is increased by a factor of 100, what happens to the pH of the solution?
17. log10 (3m ⫺ 5) ⫹ log10 m ⫽ log10 2 2
18. log10 (b ⫹ 3) ⫹ log10 b ⫽ log10 4 1
22. log4 (x2 ⫺ 4) ⫺ log4 (x ⫹ 2) ⫽ log4 1 3
23. log10 4 ⫹ log10 w ⫽ 2 25
24. log8 (n ⫺ 3) ⫹ log8 (n ⫹ 4) ⫽ 1 4
25. 3 log5 (x2 ⫹ 9) ⫺ 6 ⫽ 0 ⫾4
26. log16 (9x ⫹ 5) ⫺ log16 (x2 ⫺ 1) ⫽ ᎏ 3
27. log6 (2x ⫺ 5) ⫹ 1 ⫽ log6 (7x ⫹ 10) 8
28. log2 (5y ⫹ 2) ⫺ 1 ⫽ log2 (1 ⫺ 2y) 0
1 2
30. log7 x ⫹ 2 log7 x ⫺ log7 3 ⫽ log7 72 6
31. SOUND Recall that the loudness L of a sound in decibels is given by L ⫽ 10 log10 R, where R is the sound’s relative intensity. If the intensity of a certain sound is tripled, by how many decibels does the sound increase? about 4.8 db
log2 a ⫹ log2 b ⫽ log2 (a ⫹ b).
Adjective tiny small
0ⱕS⬍1
medium
1ⱕS⬍2
large
2ⱕS⬍3
big
3ⱕS⬍4
huge
5. Derive an expression for S applied to a cube in terms of ᐉ where ᐉ is the side length of a cube.
However, after substituting the values for a and b in his problem, he amazingly still gets the right answer! The value of a was 11. What must the value of b have been?
log3 ᐍ
6. How many cubes, each one foot on a side, would have to be put together to get an object that Alicia would call “big”?
1.1
729
4. LENGTHS Charles has two poles. One pole has length equal to log7 21 and the other has length equal to log7 25. Express the length of both poles joined end to end as the logarithm of a single number.
7. How likely is it that a large object attached to a big object would result in a huge object, according to Alicia’s scale.
Not very likely; most likely the result will be big, not huge.
log7 525
Glencoe Algebra 2
32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people, and an earthquake rated at 4.5 may cause local damage. The Richter scale magnitude reading m is given by m ⫽ log10 x, where x represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that measures 4.5 on the Richter scale than one that measures 3.5? 10 times Chapter 9
24
Glencoe Algebra 2
Chapter 9
Answers
25
Glencoe Algebra 2
(Lesson 9-3)
20. log3 (a ⫹ 3) ⫹ log3 (a ⫹ 2) ⫽ log3 6 0
21. log10 (r ⫹ 4) ⫺ log10 r ⫽ log10 (r ⫹ 1) 2
⫺1 ⱕ S ⬍ 0
3. LUCKY MATH Frank is solving a problem involving logarithms. He does everything correctly except for one thing. He mistakenly writes Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A11
16. log2 d ⫽ 5 log2 2 ⫺ log2 8 4
⫺ 1) ⫺ 2 ⫽ log10 (c ⫹ 1) 101
S satisfies ⫺2 ⱕ S ⬍ ⫺1
The pH decreases by 2.
1 1 15. log3 y ⫽ ⫺log3 16 ⫹ ᎏ log3 64 ᎏ 3 4
29. log10
Then use the following table to find the appropriate adjective.
Answers
12. log8 48 ⫺ log8 w ⫽ log8 4 12
(c2
1 ᎏ log3 V, where V is volume in cubic feet. 3
⫺log10 C,
11. log6 x ⫹ log6 9 ⫽ log6 54 6
19. log8 (t ⫹ 10) ⫺ log8 (t ⫺ 1) ⫽ log8 12 2
Alicia wanted to try to quantify the terms puny, tiny, small, medium, large, big, huge, and humongous. She picked a number of objects and classified them with these adjectives of size. She noticed that the scale seemed exponential. Therefore, she came up with the following definition. Define S to be
2. POWERS A chemist is formulating an acid. The pH of a solution is given by
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 3
following information.
1.113
25 7
Solve each equation. Check your solutions. 9. log7 n ⫽ ᎏ log7 8 4
SIZE For Exercises 5-7, use the
Lesson 9-3
Chapter 9
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