06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 25
Home
Quit
Lesson 5.5 Exercises, pages 393–398
A
4. Simplify each expression. Use a calculator to verify the answer.
a) log 6 ⫹ log 5
ⴝ log (6 # 5)
ⴝ log 30
Verify: log 6 ⴙ log 5
ⴝ 1.4771. . .
log 30 ⴝ 1.4771. . .
c) log 48 ⫺ log 6
ⴝ log a b
48
6
ⴝ log 8
Verify: log 48 ⴚ log 6
ⴝ 0.9030. . .
log 8 ⴝ 0.9030. . .
©P DO NOT COPY.
b) 3 log 2
ⴝ log 23
ⴝ log 8
Verify: 3 log 2 ⴝ 0.9030. . .
log 8 ⴝ 0.9030. . .
d) log 8 ⫹ log 5
ⴝ log (8 # 5)
ⴝ log 40
Verify: log 8 ⴙ log 5 ⴝ 1.6020. . .
log 40 ⴝ 1.6020. . .
5.5 The Laws of Logarithms—Solutions
25
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 26
Home
Quit
5. Write each expression as a single logarithm.
a) log a ⫺ log b
b) log x ⫹ log y
ⴝ log a b
ⴝ log xy
a
b
d) log x ⫺ log y ⫹ log z
c) 5 log m
ⴝ log a yb ⴙ log z
ⴝ log m5
x
ⴝ log a y b
xz
B
6. Use each law of logarithms to write an expression that is equal to
log 16. Use a calculator to verify each expression.
a) the product law
Sample response: Determine 2 numbers whose product is 16: 2 and 8
So, log 16 ⴝ log 2 ⴙ log 8
Verify: log 16 ⴝ 1.2041. . .
log 2 ⴙ log 8 ⴝ 1.2041. . .
b) the quotient law
Determine 2 numbers whose quotient is 16: 80 and 5
So, log 16 ⴝ log 80 ⴚ log 5
Verify: log 16 ⴝ 1.2041. . .
log 80 ⴚ log 5 ⴝ 1.2041. . .
c) the power law
Write 16 as a power: 42
So, log 16 ⴝ 2 log 4
Verify: log 16 ⴝ 1.2041. . .
2 log 4 ⴝ 1.2041. . .
7. Substitute values of a and b to verify each statement.
log a
a
Z log a b
a)
b) log (a ⫹ b) Z log ab
log b
b
Substitute: a ⴝ 3 and b ⴝ 5
log a
log b
ⴝ
log 3
log 5
ⴝ 0.6826. . .
log a b ⴝ log a b
a
b
3
5
ⴝ ⴚ0.2218. . .
Since the left side is not equal
to the right side, the
statement is verified.
26
Substitute: a ⴝ 3 and b ⴝ 5
log (a ⴙ b) ⴝ log 8
ⴝ 0.9030. . .
log ab ⴝ log 15
ⴝ 1.1760. . .
Since the left side is not equal to
the right side, the statement is
verified.
5.5 The Laws of Logarithms—Solutions
DO NOT COPY. ©P
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 27
Home
Quit
8. Write each expression as a single logarithm.
1
a) log x ⫺ 5 log y
b) 2 log x ⫹ 3 log y
1
ⴝ log x ⴚ log y5
ⴝ log x2 ⴙ log y3
ⴝ log Ax2y3B
ⴝ log a 5b
1
x
y
2
c) 3 log5x ⫺ 4 log5y ⫺ 3 log5z
2
ⴝ log5x3 ⴚ log5y4 ⴚ log5z3
2
ⴝ log5x3 ⴚ (log5y4 ⴙ log5z3)
2
ⴝ log5x3 ⴚ log5y4z3
d) 5 ⫹ log2x
ⴝ log225 ⴙ log2x
ⴝ log232 ⴙ log2x
ⴝ log232x
2
x3
ⴝ log5 ¢ 4 3≤
yz
9. Explain each step in this proof of the power law for logarithms.
To prove that logbxk ⫽ k logbx:
Let: logbx ⫽ n
Then x ⫽ bn
xk ⫽ 1bn2k
xk ⫽ bkn
logbxk ⫽ logbbkn
logbxk ⫽ kn
logbxk ⫽ k logbx
Write the logarithm as a power.
Raise each side to the power k.
Simplify.
Take the logarithm base b of each side.
Simplify the right side.
Substitute: n ⴝ logbx
10. Use the strategy from the proof of the product law for logarithms to
x
prove the quotient law: logb ay b ⫽ logbx ⫺ logby
Let logbx ⴝ m and logby ⴝ n
Then x ⴝ bm
x
So, y ⴝ
m
Apply the definition of a
logarithm.
y ⴝ bn
b
bn
x
mⴚn
y ⴝb
logb a yb ⴝ m ⴚ n
x
Use the quotient law for exponents.
Write this exponential statement as a
logarithmic statement.
Substitute for m and n.
logb a yb ⴝ logbx ⴚ logby
x
©P DO NOT COPY.
5.5 The Laws of Logarithms—Solutions
27
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 28
Home
Quit
11. Use two different strategies to write 2(log x ⫹ log y) as a single
logarithm.
Strategy 1
2(log x ⴙ log y)
ⴝ 2 log x ⴙ 2 log y
ⴝ log x2 ⴙ log y2
ⴝ log x2y2
Strategy 2
2(log x ⴙ log y)
ⴝ 2 log xy
ⴝ log (xy)2
ⴝ log x2y2
12. Write each expression as a single logarithm.
1
a) 3 log 2 ⫹ log 6
b) 2 log 9 ⫹ 2 log 5
ⴝ
ⴝ
ⴝ
ⴝ
log 23 ⴙ log 6
log 8 ⴙ log 6
log (8 ⴢ 6)
log 48
c) 3 log26 ⫺ 2
1
ⴝ log 92 ⴙ log 52
ⴝ log 3 ⴙ log 25
ⴝ log 75
d) 5 log52 ⫺ log54 ⫹ 2
ⴝ log263 ⴚ log222
ⴝ log2216 ⴚ log24
ⴝ log525 ⴚ log54 ⴙ log552
ⴝ log532 ⴚ log54 ⴙ log525
ⴝ log2 a
ⴝ log5 a
ⴝ log254
ⴝ log5200
216
b
4
32 # 25
b
4
13. Evaluate each expression.
b) 2 log236 ⫹ log212 ⫺ 2 log23
ⴝ log362 ⴚ log323 ⴙ log318
ⴝ log336 ⴚ log38 ⴙ log318
ⴝ log2362 ⴙ log212 ⴚ log232
ⴝ log26 ⴙ log212 ⴚ log29
ⴝ log3 a
ⴝ log2 a
ⴝ log381
ⴝ log334
ⴝ4
ⴝ log28
ⴝ log223
ⴝ3
36 # 18
b
8
c) 9 log93 ⫺ log975 ⫹ 2 log95
1
6 # 12
b
9
d) log498 ⫺ 2 log47 ⫺ 2
ⴝ log939 ⴚ log975 ⴙ log952
ⴝ log919 683 ⴚ log975 ⴙ log925
ⴝ log498 ⴚ log472 ⴚ 2
ⴝ log498 ⴚ log449 ⴚ 2
ⴝ log9 a
ⴝ log4 a b ⴚ 2
ⴝ log96561
ⴝ log994
ⴝ4
ⴝ log42 ⴚ 2
19 683 # 25
b
75
28
1
a) 2 log36 ⫺ 3 log32 ⫹ log318
98
49
1
ⴝ log442 ⴚ 2
1
ⴚ2
2
3
ⴝⴚ
2
ⴝ
5.5 The Laws of Logarithms—Solutions
DO NOT COPY. ©P
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 29
Home
Quit
14. Given log a ⬟ 1.301, determine an approximate value for each
logarithm.
a) log a3
c) log a100b
a2
b) log 10a
ⴝ 3 log a
⬟ 3(1.301)
⬟ 3.903
ⴝ log 10 ⴙ log a
⬟ 1 ⴙ 1.301
⬟ 2.301
ⴝ
ⴝ
⬟
⬟
log a2 ⴚ log 100
2 log a ⴚ 2
2(1.301) ⴚ 2
0.602
15. Identify the errors in the solution to the question below.
Write the correct solution.
1
a2
Write log ¢ 3 2≤ in terms of log a, log b, and log c.
cb
1
a2
log ¢ 3 2≤
cb
log a
1
⫽ log a2 ⫺ log c3b2
1
ⴝ log a2 ⴚ log c3b2
1
⫽ log a2 ⫺ log c3 ⫹ log b2
1
a2
b
c3b2
1
ⴝ log a2 ⴚ (log c3 ⴙ log b2)
1
⫽ 2 log a ⫺ log 3c ⫹ 2 log b ⴝ log a12 ⴚ log c3 ⴚ log b2
ⴝ
1
log a ⴚ 3 log c ⴚ 2 log b
2
In the second line of the solution, when log c3b2 is written as a sum of
logarithms, both logarithms should be negative. In the third line of the
solution, log c3 should be 3 log c.
16. Write each expression in terms of log a, log b, and/or log c.
1
2
a) log a3b2
b) log ab2c3
1
ⴝ log a3 ⴙ log b2
ⴝ 3 log a ⴙ
1
log b
2
a3
c) log a 2 b
b
ⴝ log a3 ⴚ log b2
ⴝ 3 log a ⴚ 2 log b
©P DO NOT COPY.
2
ⴝ log a ⴙ log b2 ⴙ log c3
ⴝ log a ⴙ 2 log b ⴙ
2
log c
3
3
a4b5
d) log ¢
≤
c
3
ⴝ log a4 ⴙ log b5 ⴚ log c
ⴝ 4 log a ⴙ
3
log b ⴚ log c
5
5.5 The Laws of Logarithms—Solutions
29
06_ch05_pre-calculas12_wncp_solution.qxd
5/22/12
4:03 PM
Page 30
Home
Quit
17. Given log 3 ⬟ 0.477 and log 7 ⬟ 0.845, determine the approximate
value of log (132 300) without using a calculator.
Write 132 300 as a product of factors that involve powers of 3 and 7.
132 300 ÷ 9 ⴝ 14 700
14 700 ÷ 3 ⴝ 4900
4900 ÷ 49 ⴝ 100
So, 132 300 ⴝ 33 ⴢ 72 ⴢ 102
log (132 300) ⴝ log (33 ⴢ 72 ⴢ 102)
ⴝ log 33 ⴙ log 72 ⴙ log 102
ⴝ 3 log 3 ⴙ 2 log 7 ⴙ 2
⬟ 3(0.477) ⴙ 2(0.845) ⴙ 2
⬟ 5.121
C
18. Write each expression as a single logarithm.
a) 3 log x ⫹ log (2x ⫺ 3)
ⴝ log x3 ⴙ log (2x ⴚ 3)
ⴝ log x3(2x ⴚ 3)
b) log (x ⫹ 1) ⫹ log (2x ⫺ 1)
ⴝ log (x ⴙ 1)(2x ⴚ 1)
c) log (x2 ⫺ 1) ⫺ log (x ⫺ 1) d) log (2x2 ⫹ x ⫺ 3) ⫺ log (x2 ⫺ 1)
x2 ⴚ 1
b
xⴚ1
ⴝ log a
ⴝ log
2x2 ⴙ x ⴚ 3
b
x2 ⴚ 1
(2x ⴙ 3)(x ⴚ 1)
ⴝ log a
(x ⴚ 1)(x ⴙ 1)
xⴚ1
ⴝ log (x ⴙ 1)
ⴝ log
(x ⴚ 1)(x ⴙ 1)
2x ⴙ 3
b
xⴙ1
ⴝ log a
19. Without using the power law, prove the law of logarithms for radicals:
√
1
logb k x ⫽ logbx, b > 0, b Z 1, k H ⺞, x > 0
k
Sample response:
Let: logbx ⴝ n
Then
Write the logarithm as a power.
1
k
xⴝb
n
1
Raise each side to the power .
n
xk ⴝ bk
1
k
Take the logarithm base b of each side.
n
k
logb x ⴝ logbb
n
logb x ⴝ
k
1
1
logb xk ⴝ logbx
k
√
1
Or, logb k x ⴝ logbx
k
1
k
30
Simplify the right side.
Substitute: n ⴝ logbx
5.5 The Laws of Logarithms—Solutions
DO NOT COPY. ©P