Name: __________________________ Date: _____________
1. Identify the graph of the function.
2. Sketch the graph of the function
.
3. Identify the operation that will transform the graph of f ( x)  2 x into the graph of
g ( x)  3  2 x .
4. Sketch the graph of the function
.
5.
1
 
3
Solve the following equation for x.
7 x1
 27
6. Let Q represent a mass of radioactive radium (226Ra) (in grams), whose half-life is 1599
years. The quantity of radium present after t years is
t /1599
1
Q  4 
2
Determine the quantity present after 100 years. Round to the nearest hundredth of a
gram.
7.
Rewrite the logarithmic equation log 2
8.
Rewrite the exponential equation 4 –2 
1
 – 2 in exponential form.
4
1
in logarithmic form.
16
9. Identify the x-intercept of the function y  3  log 2 x .
10. Identify the vertical asymptote of the function f ( x)  1  log( x  4) .
11. Graph the function.
12. Write the logarithmic equation ln 5  1.609... in exponential form.
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13. Write the exponential equation e3/2  4.4817... in logarithmic form.
14. Identify the x-intercept of the function f ( x)  2ln( x  4) .
15. Solve the equation ln x  ln(e4 ) for x.
16. Rewrite the logarithm log5 33 in terms of the natural logarithm.
17. Evaluate the logarithm log1/3 0.995 using the change of base formula. Round to 3
decimal places.
18.
4
1
Simplify the expression log3   .
9
19. Find the exact value of log 4 28  log 4 7 without using a calculator.
20. Find the exact value of ln e4.50  ln e without using a calculator.
21. Find the exact value of log 3 49 without using a calculator.
7
22. Expand the expression as a sum, difference, and/or constant multiple of logarithms.
23.
24.
Condense the expression
1
[log3 x  log3 5]  [log3 y] to the logarithm of a single term.
3
Put the expressions in the appropriate order:
ln 3 ln 3
,
, ln 3 .
ln e e
25. The pH of an acidic solution is a measure of the concentration of the acid particles in the
solution, with smaller values of the pH indicating higher acid concentration. In a lab
experiment, the pH of a certain acid solution is changed by dissolving over-the-counter
antacid tablets into the solution. In this experiment, the pH changes according to the
equation
x 

pH  4.67  log 
,
 0.25  x 
where x is the number of grams of antacid added to the solution. Use the properties of
logarithms to write the formula involving sums and/or differences of logarithms.
26.
1
Determine whether or not x  (e2  1) is a solution to ln(2 x  1)   2 .
2
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27.
x
1
Solve    64 for x.
4
28. Solve ln x2  ln 7  0 for x.
29. Solve for x: 3 x /3  0.0052 . Round to 3 decimal places.
30. Solve for x: e x (2  e x )  1. Round to 3 decimal places.
31. Approximate the solution to ln( x  2)  ln x  3 . Round to 3 decimal places.
32. Solve log6 x  log6 (2 x  1)  2 . Round to 3 decimal places.
33. Glove and mitten warmers have become popular over the past few years. These devices
consist of a paper pouch filled with a chemical mixture that reacts and releases heat
when exposed to air. At a time t (in minutes) after being exposed to air at a temperature
of 32 oF, the temperature T of the pouch (in oF) can be modeled by the equation
2
T (t )  88e11/t  32, 0  t  200
Find the time necessary for the warmer to reach a temperature of 108 oF. Round to the
nearest tenth of a minute.
34. Identify the graph that represents the function.
35. An initial investment of $2000 grows at an annual interest rate of 5% compounded
continuously. How long will it take to double the investment?
36. An initial investment of $2000 doubles in value in 11 years. Assuming continuous
compounding, what was the interest rate? Round to the nearest tenth of a percent.
37. How long will it take an investment that pays 4% compounded annually to double in
value? Round to the nearest tenth of a year.
38. A hunting club stocks a wildlife preserve with 18 elk. The carrying capacity of the
preserve is 234 animals and the growth of the herd, allowing for the effect of controlled
hunting of the elk, is expected to be modeled by the logistic curve
234
,
p t  
1  12e –0.152t
where t is measured in years. After how many years will the population be 212? Round
to the nearest year.
39. What is the half-life of a radioactive substance if 3.9 g decays to 0.80 g in 45 hours?
Round to the nearest tenth of an hour.
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40. The population P of a bacteria culture is modeled by P  4300ekt , where t is the time in
hours. If the population of the culture was 5800 after 40 hours, how long does it take for
the population to double? Round to the nearest tenth of an hour.
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Answer Key
1.
2.
3. g(x) is obtained by reflecting f(x) in the x-axis then shifting upward 3 units.
4.
5.
2
7
6. 3.83 g
7. –2 1
2 
4

Page 5
8.
log 4
1
 –2
16
9. 1
8
10. x   4
11.
12. e1.609...  5
13.
3
ln  4.4817... 
2
14. x = 5
15. e 4
16. ln 33
ln 5
17. 0.005
18. –8
19. 1
20. 4
21. 2
3
22.
23.
log3
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
3 5x
y
ln 3 ln 3

 ln 3
e
ln e
pH  4.67  log x  log  0.25  x 
yes
3
 7, 7
14.361
0.000
0.105
4.000
8.7 minutes
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35.
36.
37.
38.
39.
40.
13.86 years
6.3%
17.7 years
31 years
19.7 hours
92.7 hours
Page 7