3-3 Properties of Logarithmsmsgalgebra2preap.weebly.com/uploads/1/3/3/8/13386902/3.3... · 2018. 9....
12
Express each logarithm in terms of ln 2 and ln 5. 1. ln SOLUTION: 3. ln 80 SOLUTION: 5. ln SOLUTION: 7. ln 2000 SOLUTION: Express each logarithm in terms of ln 3 and ln 7. 9. ln 63 SOLUTION: eSolutions Manual - Powered by Cognero Page 1 3-3 Properties of Logarithms
Transcript of 3-3 Properties of Logarithmsmsgalgebra2preap.weebly.com/uploads/1/3/3/8/13386902/3.3... · 2018. 9....
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 1
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 2
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 3
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 4
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 5
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 6
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 7
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 8
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 9
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 10
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 11
3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln 5.
1. ln
SOLUTION:
3. ln 80
SOLUTION:
5. ln
SOLUTION:
7. ln 2000
SOLUTION:
Express each logarithm in terms of ln 3 and ln 7.9. ln 63
SOLUTION:
11. ln
SOLUTION:
13. ln 1323
SOLUTION:
15. ln
SOLUTION:
18. TORNADOES The distance d in miles that a tornado travels is , where w is the wind speed in miles perhour of the tornado. a. Express w in terms of log d. b. If a tornado travels 100 miles, estimate the wind speed.
SOLUTION: a.
b.
Evaluate each logarithm.
21. 9 ln e3 + 4 ln e
5
SOLUTION:
24.
SOLUTION:
27.
SOLUTION:
Expand each expression.
30.
SOLUTION:
33. log11 ab−4
c12
d7
SOLUTION:
36. log5 a6b−3
c4
SOLUTION:
Condense each expression.
39.
SOLUTION:
42.
SOLUTION:
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
48.
SOLUTION:
Evaluate each logarithm.51. log7 5
SOLUTION:
54. log22 400
SOLUTION:
57. log−2 8
SOLUTION: no real solution
60. TRUCKING Bill’s Trucking Service purchased a new delivery truck for $56,000. Suppose t = log(1 – r)
represents the time t in years that has passed since the purchase given its initial price P, present value V, and
annual rate of depreciation r. a. If the truck's present value is $40,000 and it has depreciated at a rate of 15% per year, how much time has passed since its purchase to the nearest year? b. If the truck's present value is $34,000 and it has depreciated at a rate of 10% per year, how much time has passed since its purchase to the nearest year?
SOLUTION: a.
about 2 years b.
about 5 years
Estimate each logarithm to the nearest whole number.63. log3 10
SOLUTION:
66. log12 177
SOLUTION:
Expand each expression.
69.
SOLUTION:
72.
SOLUTION:
75. EARTHQUAKES The Richter scale measures the intensity of an earthquake. The magnitude M of the seismic
energy in joules E released by an earthquake can be calculated by .
a. Use the properties of logarithms to expand the equation.
b. What magnitude would an earthquake releasing 7.94 × 1011
joules have?
c. The 2007 Alum Rock earthquake in California released 4.47 × 1012 joules of energy. The 1964 Anchorage
earthquake in Alaska measured a magnitude of 1.58 × 1018
joules of energy. How many times as great was the magnitude of the Anchorage earthquake as the magnitude of the Alum Rock earthquake? d. Generally, earthquakes cannot be felt until they reach a magnitude of 3 on the Richter scale. How many joules of energy does an earthquake of this magnitude release?
SOLUTION: a.
b.
c.
Condense each expression.
78.
SOLUTION:
81. ln x + ln (y + 8) – 3 ln y – ln (10 – x)
SOLUTION:
Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
84. ln 162
SOLUTION:
87. ln
SOLUTION:
Determine the graph that corresponds to each equation.
90. f (x) = ln x + ln (x + 3)
SOLUTION:
Make a table of values.
This table resembles the graphs for a, c, and d. However, the origin is a point on the graph of d. f (x) is undefined forx = 1 or x = 2 in graph c. The correct choice is a.
x f(x) 0 undefined 1 1.39 2 2.30 3 2.89 4 3.33
93. f (x) = 0.5 ln (x – 2)
SOLUTION: Make a table of values.
f(x) is undefined for x ≤ 2. This table resembles graph c.
x f(x) 2 undefined 3 0 4 0.35 5 0.55 6 0.69
eSolutions Manual - Powered by Cognero Page 12
3-3 Properties of Logarithms
![based ionic liquids: OFET application and tribological ... · 3 [P22P113N][NTf2]4.3H2O (PzIL2a) 1 35-292 188 2.89 2.82 3H2O C39H73F25N13O23P3S8 2 292-454 371 87.78 1916.44 gmol-1](https://static.fdocuments.net/doc/165x107/5fdb557210743856e36f0850/based-ionic-liquids-ofet-application-and-tribological-3-p22p113nntf243h2o.jpg)
