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Solve the problem.
125)
The function A =A0e-0.01386x models the amount in pounds of a particular radioactive material
stored in a concrete vault, where x is the number of years since the material was put into the vault.
If 800 pounds of the material are placed in the vault, how much time will need to pass for only 100
pounds to remain?
125)
A)
160 yr
B)
150 yr
C)
155 yr
D)
300 yr
Solve.
126)
The population of a certain country is growing at a rate of 1.4% per year. How long will it take for
this country's population to double? Use the formula t =ln 2
k, which gives the time, t, for a
population with growth rate k, to double. Round to the nearest year.
126)
A)
50 yr
B)
51 yr
C)
49 yr
D)
52 yr
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
127)
log10100x5
127)
A)
2x
B)
2+ 5 log10 x
C)
20 + 5 log10 x
D)
2log10 x
Approximate the number using a calculator. Round the answer to three decimal places.
128)
6-2.6
128)
A)
308.916
B)
0.009
C)
-15.600
D)
0.309
Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.
129)
f(x) =1
x and g(x) =x2
129)
A)
not inverses
B)
inverses
Solve.
130)
The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and
has a pH of 7. The pH of a solution is given by pH = - log x where x represents the concentration of
the hydrogen ions in the solution in moles per liter. Find the pH if the hydrogen ion concentration
is 4.3 × 10-9.
130)
A)
8.63
B)
8.37
C)
9.37
D)
9.63
131)
The value of a particular investment follows a pattern of exponential growth. You invested money
in a money market account. The value of your investment t years after your initial investment is
given by the exponential growth model A =9400e0.062t. By what percentage is the account
increasing each year?
131)
A)
9.4%
B)
6.2%
C)
6.9%
D)
0.62%
Evaluate the expression without using a calculator.
132)
log644
132)
A)
1
3
B)
3
C)
12
D)
1
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
133)
log3
y
81
133)
A)
log3y-4log33
B)
log3y-log381
C)
1
2log3 y -4
D)
1
2log3 y -log381
Approximate the number using a calculator. Round the answer to three decimal places.
134)
e-2.5
134)
A)
-0.082
B)
0.082
C)
0.382
D)
-6.796
Solve the equation by expressing each side as a power of the same base and then equating exponents.
135)
1
5
x
=1
625
135)
A)
1
4
B)
1
125
C)
{4}
D)
{-4}
Solve the exponential equation by taking the logarithm on both sides. Use a calculator to obtain a decimal approximation,
correct to four decimal places, for the solution.
136)
e3x =6
136)
A)
0.5973
B)
5.3753
C)
5.4366
D)
0.1831
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
137)
log69+log64
137)
A)
2
B)
log613
C)
2log66
D)
log636
Solve the logarithmic equation. Give an exact answer.
138)
log 2(5x - 1) =log 2(2x + 4)
138)
A)
1
B)
5
3
C)
{3]
D)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
139)
log2(4x)
139)
A)
4+log2 x
B)
2x
C)
2log2 x
D)
2+log2 x
Solve the logarithmic equation. Give an exact answer.
140)
log6(x2- x) = 1
140)
A)
{2, 3}
B)
{-2, -3}
C)
{1, 6}
D)
{-2, 3}
Find the inverse of the one-to-one function.
141)
f(x) =
6x
141)
A)
f-1(x) =x6
B)
f-1(x) =1
6x
C)
f-1(x) =6x
D)
f-1(x) =x +6
Write the equation in its equivalent logarithmic form.
142)
92= y
142)
A)
logy2=9
B)
log9y =2
C)
logy9=2
D)
log2y =9
Solve the logarithmic equation. Give an exact answer.
143)
log5(x + 2) -log5 x = 2
143)
A)
2
25
B)
12
C)
{5}
D)
1
12
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
144)
ln y-z
144)
A)
z ln y
B)
-z ln y
C)
-y ln z
D)
y ln z
Solve the problem.
145)
The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and
has a pH of 7. The pH of a solution is given by pH = - log x where x represents the concentration of
the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the
pH =6.
145)
A)
106
B)
1.79
C)
0.17
D)
10-6
Find the composition.
146)
If f(x) =6x and g(x) = x + 10, find (f
g)(4).
146)
A)
64
B)
264
C)
34
D)
84
Solve.
147)
The rabbit population in a forest area grows at the rate of 4% monthly. If there are 230 rabbits in
April, find how many rabbits (rounded to the nearest whole number) should be expected by next
April. Use the function f(x) =230(2.7)0.04t.
147)
A)
357 rabbits
B)
298 rabbits
C)
383 rabbits
D)
370 rabbits
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
148)
log
3x
1000
148)
A)
-30x
B)
1000x
C)
1
3 log x -3
D)
1
3 log x +3
Provide an appropriate response.
46
149)
Graph f(x) =4xand g(x) =4x +2 in the same rectangular coordinate system.
149)
A)
B)
C)
D)
47
Solve the logarithmic equation. Give an exact answer.
150)
log10 7+log10 x = 1
150)
A)
1
7
B)
{710}
C)
7
10
D)
10
7
Solve the problem.
151)
The first recorded population of a particular country was 24 million, and the population was
recorded as 32 million 14 years later. Write an exponential growth function that describes the
population of this country, in millions, t years after the first recorded population.
151)
A)
A =24e0.149t
B)
A =24e0.475t
C)
A =24e0.021t
D)
A =24e0.031t
Graph the function.
152)
Use the graph of f(x) =4x to obtain the graph of g(x) =4x + 3 + 1.
152)
A)
B)
48
C)
D)
Write the equation in its equivalent logarithmic form.
153)
4-3=1
64
153)
A)
log1/44= - 3
B)
log- 3
1
64 =4
C)
log4-3=1
64
D)
log4
1
64 = - 3
Solve the exponential equation by taking the logarithm on both sides. Use a calculator to obtain a decimal approximation,
correct to four decimal places, for the solution.
154)
4(x +6) =3
154)
A)
-0.9031
B)
7.2619
C)
-5.2075
D)
2.0400
Provide an appropriate response.
155)
Find the domain: f(x) =log3(x -7).
155)
A)
(7, )
B)
(-3, )
C)
(3, )
D)
(-7, )
Write the equation in its equivalent logarithmic form.
156)
23= x
156)
A)
log2x =3
B)
logx2=3
C)
log23= x
D)
log3x =2
Find the composition.
157)
If f(x) =2x +5 and g(x) = x +3, find (g
f)(x).
157)
A)
2x +11
B)
2x +8
C)
2x2+11x +15
D)
2x2+11
Solve the problem.
158)
A research team collected data concerning a local welfare-to-work program. They found that the
more frequently a participant attended weekly group support meetings, the longer he or she would
keep the same job. The data are summarized in the graph below. Call the function f.
Does f have an inverse? Why or why not?
158)
A)
No, because it fails the horizontal line test.
B)
No, because it fails the vertical line test.
C)
Yes, because it passes the vertical line test.
D)
Yes, because it passes the horizontal line test.
Evaluate the expression without using a calculator.
159)
ln e6
159)
A)
1
6
B)
6
C)
-6
D)
log 6
Solve the logarithmic equation. Give an exact answer.
160)
log 3x =4
160)
A)
{64}
B)
{81}
C)
ln 4
ln 3
D)
{12}
Solve the equation.
161)
4(3x - 1) =1024
161)
A)
{4}
B)
{2}
C)
{3}
D)
{1}
162)
log2(x + 3) = - 3
162)
A)
{25
9}
B)
25
8
C)
-23
8
D)
{-23
9}
Approximate the number using a calculator. Round the answer to three decimal places.
163)
e3.5
163)
A)
33.415
B)
33.115
C)
30.125
D)
9.514
Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.
164)
f(x) =3
x +5 and g(x) =5
x-3
164)
A)
inverses
B)
not inverses
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
165)
4 ln(x -12) -5 ln x
165)
A)
ln 4(x -12)
5x
B)
ln(x5(x -12)4)
C)
ln (x -12)4
x5
D)
ln(20x(x -12))
Graph the function by making a table of coordinates.
166)
f(x) =5
3
x
166)
A)
B)
52
C)
D)
Solve the problem.
167)
The formula y = 1 +1.3 ln(x + 1) models the average number of free-throws a basketball player can
make consecutively during practice as a function of time, where x is the number of consecutive
days the basketball player has practiced for two hours. After how many days of practice can the
basketball player make an average of 10 consecutive free throws?
167)
A)
4728 days
B)
1014 days
C)
4730 days
D)
1016 days
168)
The regular price of a car is x dollars. Let f(x) = x -600 and g(x) =0.95x. Then f represents the price
of the car less $600, and g represents a 5% discount on the car. Which composite function represents
the greater discount on the car, f
g or g
f?
168)
A)
f
g and g
f don't represent discounts on the car.
B)
f
g represents the greater discount.
C)
g
f represents the greater discount.
D)
f
g and g
f represent an equal discount.
Solve the logarithmic equation. Give an exact answer.
169)
log(4+ x) - log(x - 2) = log 3
169)
A)
B)
1
2
C)
{-5}
D)
{5}
Determine if the graph represents a function that has an inverse function.
170)
170)
A)
does not have an inverse
B)
has an inverse
Solve the problem.
171)
Cindy will require $20,000 in 3 years to return to college to get an MBA degree. How much money
should she ask her parents for now so that, if she invests it at 12% compounded continuously, she
will have enough for school? (Round your answer to the nearest dollar.)
171)
A)
$14,236
B)
$9735
C)
$13,954
D)
$28,667
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
172)
4 ln x -1
3 ln y
172)
A)
ln(x43y)
B)
ln(x4y3)
C)
ln x4
y3
D)
ln x4
3y
Find the inverse of the one-to-one function.
173)
f(x) =3x +11
x +5
173)
A)
f-1(x) =3x -11
x -5
B)
f-1(x) =5x -11
3- x
C)
f-1(x) =x2+5x -11
3
D)
f-1(x) =x +5
3x +11
Evaluate the expression without using a calculator.
174)
log66
174)
A)
1
B)
0
C)
1
6
D)
6
Approximate the number using a calculator. Round the answer to three decimal places.
175)
22.1
175)
A)
4.200
B)
4.587
C)
4.287
D)
4.410
Provide an appropriate response.
176)
If f(x) =7x - 7, find f-1(x).
176)
A)
f-1(x) =7x + 7
7
B)
f-1(x) =y + 7
7
C)
f-1(x) =x - 7
7
D)
f-1(x) =x + 7
7
Graph the function.
177)
Use the graph of f(x) =4x to obtain the graph of g(x) =4x- 1.
177)
A)
B)
C)
D)
178)
f(x) =log1/4x
178)
A)
B)
C)
D)
Solve the equation.
179)
log7 x +log7(4x -3) = 1
179)
A)
-1, 7
4
B)
{-1}
C)
2
D)
7
4
Provide an appropriate response.
180)
Use a calculator to evaluate log63 to four decimal places.
180)
A)
0.6131
B)
1.6309
C)
-0.3010
D)
1.2553
Solve the exponential equation by taking the logarithm on both sides. Use a calculator to obtain a decimal approximation,
correct to four decimal places, for the solution.
181)
e(x +3) =6
181)
A)
-0.8028
B)
-1.2082
C)
2.1972
D)
-1.1282
Graph the function.
182)
Use the graph of f(x) =4x to obtain the graph of g(x) =4x - 3.
182)
58
A)
B)
C)
D)
Solve the problem.
183)
The function A =259e0.039t models the population of a particular city, in thousands, t years after
2011. When will the population of the city reach 368 thousand? Is the population increasing or
decreasing. Explain.
183)
A)
2019; Decreasing; The growth rate, -0.039, is negative
B)
2021; Increasing; The growth rate, 0.039, is positive.
C)
2020; Increasing; The growth rate, 0.039, is positive.
D)
2022; Decreasing; The growth rate, -0.039, is negative
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose
coefficient is 1. Where possible, evaluate logarithmic expressions.
184)
3 ln a -5 ln b
184)
A)
ln a3
b5
B)
ln a
b
8
C)
ln a3
ln b5
D)
ln 3a
5b
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
185)
log x
10
185)
A)
10x
B)
log x -1
C)
log x +1
D)
-10x
186)
log3
81
x - 1
186)
A)
4-1
2log3(x - 1)
B)
log381 -log3x - 1
C)
4-log3x - 1
D)
4log33-1
2log3(x - 1)
Graph the function.
187)
Use the graph of f(x) =3x to obtain the graph of g(x) =3-x.
187)
