Name: ______________________
Class: _________________
Date: _________
ID: A
Unit 3 Test Review: Logarithms and Inverse Functions
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
1. Graph y = −4x 2 − 2 and its inverse.
a.
c.
b.
d.
2. Kepler’s Third Law of Orbital Motion states that you can approximate the period P (in Earth years) it
3
2
takes a planet to complete one orbit of the sun using the function P = d , where d is the distance (in
astronomical units, AU) from the planet to the sun. How many Earth years would it take for a planet
that is 6.76 AU from the sun?
a. 15.23
c. 154.46
b. 17.58
d. 3.58
1
What is the inverse of the given relation?
____
____
3. y = 7x 2 − 3.
a.
y = ±
b.
x =
____
____
____
____
y+3
7
4. y = 3x + 9
1
a. y = x + 3
3
x−3
7
c.
y2 =
d.
y = ±
c.
y = 3x + 3
x−3
7
1
x−3
3
5. Police can estimate the speed of a vehicle before the brakes are applied using the formula
s2
0.75d =
, where s is the speed in miles per hour and d is the length of the vehicle’s skid marks in
30.25
feet. What was the approximate speed of a vehicle that left a skid mark measuring 100 feet?
a. about 29 miles per hour
c. about 48 miles per hour
b. about 10 miles per hour
d. about 43 miles per hour
b.
____
x+3
7
y = 3x − 3
d.
y=
2
6. The function d = 4.9t represents the distance d, in meters, that an object falls in t seconds due to
Earth’s gravity. Find the inverse of this function. How long, in seconds, does it take for a cliff diver
who is 70 meters above the water to reach the water below?
a. 3.8 seconds
c. 8.1 seconds
b. 5.9 seconds
d. 13.8 seconds
7. Find the annual percent increase or decrease that y = 0.35(2.3) x models.
a. 230% increase
c. 30% decrease
b. 130% increase
d. 65% decrease
8. An initial population of 820 quail increases at an annual rate of 23%. Write an exponential function to
model the quail population. What will the approximate population be after 3 years?
x
a. f(x) = 820(1.23) x ; 1526
c. f(x) = ( 820 ⋅ 0.23 ) ; 6,708,494
b. f(x) = 820(23) x ; 9,976,940
d. f(x) = 820(0.23) x ; 1526
9. The half-life of a certain radioactive material is 32 days. An initial amount of the material has a mass of
361 kg. Write an exponential function that models the decay of this material. Find how much
radioactive material remains after 5 days. Round your answer to the nearest thousandth.
a.
ÁÊ 1
y = 361 ÁÁÁÁ
ÁË 2
32 x
˜ˆ˜
˜˜ ; 0 kg
˜˜
¯
b.
ÊÁ 1
y = 361 ÁÁÁÁ
ÁË 2
ˆ˜
˜˜
˜˜
˜¯
1
32
c.
ÁÊ 1
y = 2 ÁÁÁÁ
ÁË 361
d.
1 ÊÁÁÁ 1
y =
ÁÁ
2 ÁË 361
x
; 323.945 kg
2
˜ˆ˜
˜˜
˜˜
¯
1
32
ˆ˜
˜˜
˜˜
˜¯
x
; 0.797 kg
1
32
x
; 0.199 kg
____ 10. The half-life of a certain radioactive material is 71 hours. An initial amount of the material has a mass
of 722 kg. Write an exponential function that models the decay of this material. Find how much
radioactive material remains after 17 hours. Round your answer to the nearest thousandth.
a.
1
y =
2
ÊÁ 1
ÁÁ
ÁÁ
ÁË 722
ˆ˜
˜˜
˜˜
˜¯
1
71
1
x
; 0.103 kg
c.
ÊÁ 1
y = 2 ÁÁÁÁ
ÁË 722
ˆ˜
˜˜
˜˜
˜¯
1
71
x
; 0.414 kg
x
71 x
ÁÊ 1 ˜ˆ
y = 722 ÁÁÁÁ ˜˜˜˜ ; 0 kg
ÁË 2 ˜¯
____ 11. Suppose you invest $1600 at an annual interest rate of 4.6% compounded continuously. How much will
you have in the account after 4 years?
a. $800.26
b. $6,701.28
c. $10,138.07
d. $1,923.23
____ 12. How much money invested at 5% compounded continuously for 3 years will yield $820?
a. $952.70
b. $818.84
c. $780.01
d. $705.78
b.
ÁÊ 1
y = 722 ÁÁÁÁ
ÁË 2
˜ˆ˜
˜˜
˜˜
¯
71
; 611.589 kg
d.
Write the equation in logarithmic form.
____ 13. 2 5 = 32
a. log 32 = 5 ⋅ 2
b. log 2 32 = 5
c.
d.
log 32 = 5
log 5 32 = 2
4
____ 14. 125
3
= 625
4
3
3
3
b. 3 log 4 625 = 125
d. log 625 125 =
4
____ 15. The table shows the location and magnitude of some notable earthquakes. How many times more energy
was released by the earthquake in Peru than by the earthquake in Mexico?
a.
log 4 625 = 125
c.
Earthquake Location
Italy
El Salvador
Afghanistan
Mexico
Peru
a.
b.
log 125 625 =
Date
October 31, 2002
February 13, 2001
May 30, 1998
January 22, 2003
June 23, 2001
about 15 times as much energy
about 5.48 times as much energy
Richter Scale Measure
5.9
6.6
6.9
7.6
8.1
c.
d.
about 0.50 times as much energy
about 37.52 times as much energy
c.
–4
Evaluate the logarithm.
____ 16. log 5
a.
1
625
–3
b.
5
3
d.
4
____ 17. log 0.01
a. –10
b.
–2
c.
2
d.
10
c.
log 7 10
d.
log 10
c.
log x(x + 2) 24
d.
none of these
Write the expression as a single logarithm.
____ 18. log 7 50 − log 7 5
a.
log 45
b.
log 7 45
____ 19. 4 log x − 6 log (x + 2)
x
a. 24 log
x+2
4
b. log x (x + 2) 6
Expand the logarithmic expression.
____ 20. log 3
d
12
log 3 d
a.
log 3 d − log 3 12
c.
b.
−d log 3 12
d.
log 3 12 − log 3 d
log 3 12
____ 21. log 3 11 p 3
a.
log 3 11 ⋅ 3 log 3 p
c.
log 3 11 + 3 log 3 p
b.
log 3 11 − 3 log 3 p
d.
11 log 3 p
____ 22. log b
3
57
74
1
1
log b 57 +
log b 74
c.
log b 57 − log b 74
2
2
1
1
1
b.
log b 57 −
log b 74
d. log b
(57 − 74)
2
2
2
____ 23. Use the Change of Base Formula to evaluate log 3 91.
a.
a. 4.106
c. 4.511
b. 1.959
d. 1.504
____ 24. Use the Change of Base Formula to evaluate log 7 40.
a. 0.527
b. 1.602
____ 25. What is the value of log8 1 3?
a. 3
1
b.
4
c.
d.
3.689
1.896
c.
4
1
3
d.
4
Solve the exponential equation.
____ 26.
1
= 64 4x − 3
16
1
a.
12
b.
1
4
c.
7
12
d.
11
12
4x
=8
3
8
3
a.
b.
c.
d. 2
4
3
8
____ 28. 125 9x − 2 = 150
a. –1.8847
b. –0.1069
c. 0.3375
d. 1.0378
____ 29. The generation time G for a particular bacteria is the time it takes for the population to double. The
t
bacteria increase in population is shown by the formula G =
, where t is the time period of the
3.3 log a P
____ 27. 4
population increase, a is the number of bacteria at the beginning of the time period, and P is the number
of bacteria at the end of the time period. If the generation time for the bacteria is 6 hours, how long will
it take 8 of these bacteria to multiply into a colony of 7681 bacteria? Round to the nearest hour.
a.
177 hours
b.
76 hours
Solve the logarithmic equation.
c.
4 hours
d.
85 hours
Round to the nearest ten-thousandth if necessary.
____ 30. 3 log 2x = 4
a. 10.7722
b. 5
c. 2.7826
d. 0.6309
____ 31. Solve log(4x + 10) = 3.
7
495
a. −
b.
c. 250
d. 990
4
2
____ 32. log(x + 9) − log x = 3
a. 0.0090
b. 0.3103
c. 3.2222
d. 111
____ 33. 2 log 4 − log 3 + 2 log x − 4 = 0
a. 12.3308
b. 43.3013
c. 86.6025
d. 1875
____ 34. Solve log 5x + log 14 = 1. Round to the nearest hundredth if necessary.
a. 28
b. 0.14
c. 3.57
d. 700
Write the expression as a single natural logarithm.
____ 35. 3 ln x − 2 ln c
x3
a. ln
b. ln ÁÊÁË x 3 + c 2 ˜ˆ˜¯
c2
1
____ 36. 3 ln a −
(ln b + ln c 2 )
2
3a
3
a
a. ln
b.
ln
2
0.5bc
2
bc 2
c.
ln ÁÊÁË x 3 − c 2 ˜ˆ˜¯
c.
ln
5
a3
bc
d.
ln x 3 c 2
d.
ln
a3
c
b
____ 37. Simplify ln e 3 .
1
c. 3e
3e
____ 38. Solve ln(2x − 1) = 8. Round to the nearest thousandth.
a. 1,489.979
b. 2,979.958
c. 2,981.458
____ 39. Solve ln 2 + ln x = 5. Round to the nearest tenth, if necessary.
a. 50,000
b. 74.2
c. 10
____ 40. Solve ln x − ln 6 = 0.
a. 6
b. 6e
c. e 6
a.
3
b.
d.
1
3
d.
1,490.979
d.
3
d.
ln 6
Use natural logarithms to solve the equation. Round to the nearest thousandth.
____ 41. 6e 4x − 2 = 3
a. –0.448
b. 0.327
c. 0.067
d. –0.046
4x
+
8
____ 42. 8e
= 15
a. –0.033
b. 0.264
c. –1.843
d. 2.157
____ 43. The sales of lawn mowers t years after a particular model is introduced is given by the function
y = 5500 ln(9t + 4), where y is the number of mowers sold. How many mowers will be sold 4 years after
a model is introduced? Round the answer to the nearest whole number.
a. 20,289 mowers
c. 8,811 mowers
b. 41,709 mowers
d. 19,713 mowers
Short Answer
x
44. Without graphing, determine whether the function y = ( 5.2 ) represents exponential growth or
exponential decay.
ÁÊ 2
45. Without graphing, determine whether the function y = 7 ÁÁÁÁ
ÁË 3
exponential decay.
x
˜ˆ˜
˜˜ represents exponential growth or
˜˜
¯
Other
46. In a particular region of a national park, there are currently 330 deer, and the population is increasing at
an annual rate of 11%.
a. Write an exponential function to model the deer population.
b. Explain what each value in the model represents.
c. Predict the number of deer that will be in the region after five years. Show your work.
6
ID: A
Unit 3 Test Review: Logarithms and Inverse Functions
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
B
B
A
D
C
A
B
A
B
B
D
D
B
C
B
C
B
C
D
A
C
B
A
D
B
C
C
C
D
A
B
A
B
B
A
D
A
D
B
A
1
ID: A
41. D
42. C
43. A
SHORT ANSWER
44. exponential growth
45. exponential decay
OTHER
46. a. y = 330(1.11) x
b. In the model, 330 represents the initial population of deer. The growth factor is represented by 1 +
0.11 or 1.11.
c. To predict the number of deer present after 5 years, substitute 5 into the function and evaluate.
y = 330 (1.11) x
function model
5
y = 330(1.11)
Substitute 5 for x.
y ≈ 556
Use a calculator.
There will be about 556 deer in the region.
2