C)
D)
200)
f(x) =3e-0.5x
A)
B)
53
C)
D)
201)
f(x) = ln(x – 1)
A)
B)
54
C)
D)
202)
f(x) =5ln x
A)
B)
55
C)
D)
203)
f(x) = ln x – 5
A)
B)
56
C)
D)
Solve the equation.
204)
e–0.1 =25
A)
–32.189
B)
–0.031
C)
D)
3.219
205)
4x=16
A)
3
B)
1
C)
D)
4
206)
47x =64
A)
343
B)
3
7
C)
D)
4 ln 4
7
207)
30x=164 (Round to the nearest hundredth.)
A)
2.70
B)
1.50
C)
D)
0.67
208)
17x=23 (Round to the nearest hundredth.)
A)
1.30
B)
1.11
C)
D)
0.90
209)
6x – 3 =28 (Round to the nearest hundredth.)
A)
3.54
B)
5.04
C)
D)
7.67
210)
91– x =19 (Round to the nearest hundredth.)
A)
1.75
B)
-0.34
C)
D)
-1.11
211)
10x + 1 =93 (Round to the nearest thousandth.)
A)
1.968
B)
2.968
C)
D)
9.437
57
212)
7x – 1 =44 (Round to the nearest thousandth.)
A)
1.945
B)
3.945
C)
D)
6.364
213)
(63x)(5x + 1) =7 (Round to the nearest thousandth.)
A)
0.048
B)
0.049
C)
D)
0.572
214)
83x =5x + 1 (Round to the nearest thousandth.)
A)
1.774
B)
0.348
C)
D)
0.774
215)
e–t=0.3 (Round to the nearest ten–thousandth.)
A)
-1.204
B)
1.3498
C)
D)
-1.3498
216)
log 3x =2
A)
9
B)
1000
C)
D)
8
217)
log 16 x =1
2
A)
0.00001526
B)
4
C)
D)
256
218)
log 8x =-3
A)
1
6561
B)
512
C)
D)
1
512
219)
log (x + 9) = 1 – log x
A)
1
B)
-1, 10
C)
D)
-10, 1
220)
ln(5x – 1) = ln 2– ln (x – 2)
A)
0, 11
5
B)
C)
D)
11
5
221)
log4 (x – 3) + log4 (x – 3) = 1
A)
10
B)
–10, 10
C)
D)
5
222)
log ( 4+ x) – log (x – 4 ) = log 3
A)
8
B)
– 8
C)
D)
58
Solve the problem.
223)
Use the formula L = 10 · log I
I0, where the loudness of a sound in decibels is determined by I, the number of
watt/m2 produced by the soundwave, and I0=10–12 watt/m2. A certain noise produces 9×10-4 watt/m2 of
power. What is the decibel level of this noise? Round to the nearest decibel.
A)
90 decibels
B)
9 decibels
C)
D)
206 decibels
224)
Use the formula L = 10 · log I
I0, where the loudness of a sound in decibels is determined by I, the number of
watt/m2 produced by the soundwave, and I0=10–12 watt/m2. What is the intensity in watt/m2 of a noise
measured at 78 decibels? Round to the nearest tenth.
A)
6.3 × 10-5 watt/m2
B)
2.4 × 1015 watt/m2
C)
7.8 × 10–10 watt/m2
D)
6.3 × 10-4 watt/m2
225)
Use the formula L = 10 · log I
I0, where the loudness of a sound in decibels is determined by I, the number of
watt/m2 produced by the soundwave, and I0=10–12 watt/m2. A certain noise measures 60 decibels. If the
intensity is multiplied by 10, how many decibels will the new noise measure?
A)
61 decibels
B)
600 decibels
C)
D)
60 decibels
Solve the problem. When appropriate, round the solution to the nearest tenth. As needed, use the formula
pH = – log H3O+.
226)
Find the pH if [H3O+] is 8.7 × 10 -5 .
A)
4.1
B)
4.5
C)
D)
5.9
227)
Find the pH if the [H3O+] = 1.0 × 10 -1 .
A)
1
B)
–13
C)
D)
13
228)
Find the pH if the [H3O+] =8.8 × 10 -4 .
A)
4.9
B)
3.9
C)
D)
3.1
229)
Find the [H3O+] if the pH =6 .
A)
.2
B)
1.8
C)
D)
10 -6
230)
Find the [H3O+] if the pH =1.4.
A)
2.5 × 10 -2
B)
4.0 × 10 -2
C)
D)
2.5 × 10 -1
59
231)
Find the [H3O+] if the pH = 2.3.
A)
2.0 ×102
B)
2.7 ×101
C)
D)
3.6 ×10–1
Solve the problem.
232)
In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per
second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05.
Find the walking speed of people in a city with a population of 379,901.
A)
12.9 ft/sec
B)
9.75 ft/sec
C)
D)
4.8 ft/sec
233)
In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per
second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05.
Find the walking speed of people in a city with a population of 1,550,667.
A)
14.25 ft/sec
B)
4.47 ft/sec
C)
D)
3.62 ft/sec
Solve.
234)
How long will it take for the population of a certain country to double if its annual growth rate is 7%? Round to
the nearest year. Use the exponential growth model P(t) =P0ekt.
A)
29 yr
B)
1 yr
C)
D)
4 yr
235)
The number of books in a small library increases according to the function B =9400e0.03t, where t is measured
in years. How many books will the library have after 6 years?
A)
11,254
B)
7000
C)
D)
16,119
236)
The population of a small country increases according to the function B =1,400,000e0.03t, where t is measured
in years. How many people will the country have after 5 years?
A)
1,977,553
B)
1,153,472
C)
D)
1,626,568
237)
How long will it take for prices in the economy to double at a 4% annual inflation rate? Use the exponential
growth model P(t) =P0ekt.(Round to the nearest year.)
A)
18 yr
B)
23 yr
C)
D)
14 yr
238)
An economist predicts that the buying power B(x) of a dollar x years from now will be given by the formula
B(x) =0.83x. How much will today’s dollar be worth in 5 years? Round the answer to the nearest cent.
A)
$0.91
B)
$3.80
C)
D)
$4.15
239)
The population of a particular city is increasing at a rate proportional to its size. It follows the function
P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 41,000, in how
many years is the population expected to be 102,500? (Round to the nearest year.)
A)
59 yr
B)
8 yr
C)
D)
3 yr
240)
How long will it take for the population of a certain country to triple if its annual growth rate is 6.2%? Round to
the nearest year. Use the exponential growth model P(t) =P0ekt.
A)
8 years
B)
1 year
C)
D)
18 years
241)
How long must $5900 be in a bank at 8% compounded annually to become $14,857.20? (Round to the nearest
year.)
A)
11 yr
B)
14 yr
C)
D)
13 yr
242)
At what interest rate must $4600 be compounded annually to equal $6916.70 after 7 yr? (Round to the nearest
percent.)
A)
6%
B)
8%
C)
D)
5%
243)
How long will it take for $1300 to grow to $45,600 at an interest rate of 8.9% if the interest is compounded
continuously? Round the number of years to the nearest hundredth.
A)
39.97 yr
B)
4.00 yr
C)
D)
4 yr
244)
How long will it take for $2900 to grow to $25,600 at an interest rate of 10.4% if the interest is compounded
quarterly? Round the number of years to the nearest hundredth.
A)
21.21 yr
B)
84.85 yr
C)
D)
85.98 yr
245)
Assume the cost of a car is $28,000. With continuous compounding in effect,find the number of years it would
take to double the cost of the car at an annual inflation rate of 2.1%. Round the answer to the nearest hundredth.
A)
33.01
B)
4.88
C)
D)
520.62
246)
Suppose the consumption of electricity grows at 4.9% per year, compounded continuously. Find the number of
years before the use of electricity has tripled. Round the answer to the nearest hundredth.
A)
2.24
B)
61.22
C)
D)
0.22
247)
The amount of particulate matter left in solution during a filtering process is given by the equation
P =800(2)–0.6n, where n is the number of filtering steps. Find the amounts left for n = 0 and n = 5. (Round to the
nearest whole number.)
A)
800, 100
B)
800, 6400
C)
D)
1600, 100
248)
In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given
time t and k is a negative constant determined by the nature of the material. An artifact is discovered at a certain
site. If it has 55% of the carbon–14 it originally contained, what is the approximate age of the artifact?
(carbon–14 decays at the rate of 0.0125% annually.) (Round to the nearest year.)
A)
4400 yr
B)
2077 yr
C)
D)
4783 yr
249)
In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given
time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope
decays at a rate of 0.2% annually. Determine the half–life of this isotope, to the nearest year.
A)
250 yr
B)
151 yr
C)
D)
347 yr
250)
The number of acres in a landfill is given by the function B =9300e-0.03t, where t is measured in years. How
many acres will the landfill have after 1 years? (Round to the nearest acre.)
A)
32,611
B)
8679
C)
D)
9025
251)
The decay of 966 mg of an isotope is given by A(t) =966e-0.027t, where t is time in years. Find the amount left
after 59 years.
A)
940
B)
191
C)
D)
196
252)
The sales of a mature product (one which has passed its peak) are given by the function S(t)= S0e–at, where t is
time in years. Find the sales after 13 years if a =0.24 and S0=76,800.
A)
1696
B)
2668
C)
D)
3391
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
253)
Explain why f(x) =2x is an exponential function but f(x) =x2 is not.
254)
Without using a calculator, explain why 2 must be greater than 8 but less than 16.
255)
Explain why 1 is excluded from being a logarithmic base.
256)
Explain why alog a5
= 5.
257)
Explain the error in the following: log 32 +log 3M =log 3(2 + M).
62
258)
Explain the error in the following: log 43y =log 43 ·log 4y.
259)
Explain the error in the following: log 68 –log 6N =log 6(8 – N).
260)
Explain how the graph of f(x) =ex could be used to graph the function g(x) = 1 + ln x.
261)
Explain how the equation log x = 1 could be solved using the graph of f(x) = log x.
262)
Explain why negative numbers do not have logarithms.