1. The graph of the exponential function f with base b approaches, but does not touch, the __________-axis. This axis, whose equation is __________, is a __________ asymptote.
A. x; y = 0; horizontal
B. x; y = 1; vertical
C. -x; y = 0; horizontal
D. x; y = -1; vertical
2. Evaluate the following expression without using a calculator.
8log8 19
A. 17
B. 38
C. 24
D. 19
3. Write the following equation in its equivalent exponential form.
log6 216 = y
A. 6y = 216
B. 6x = 216
C. 6logy = 224
D. 6xy = 232
4. Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln
5. The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds?
A. 10 grams after 10 seconds; 6 grams after 20 seconds
B. 12 grams after 10 seconds; 7 grams after 20 seconds
C. 4 grams after 10 seconds; 1 gram after 20 seconds
D. 8 grams after 10 seconds; 4 grams after 20 seconds
6. Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
3 ln x – 1/3 ln y
A. ln (x / y1/2)
B. lnx (x6 / y1/3)
C. ln (x3 / y1/3)
D. ln (x-3 / y1/4)
7. Write the following equation in its equivalent exponential form.
5 = logb 32
A. b5 = 32
B. y5 = 32
C. Blog5 = 32
D. Logb = 32
8. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
ex+1 = 1/e
A. {-3}
B. {-2}
C. {4}
D. {12}
9. Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
10. Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
11. Consider the model for exponential growth or decay given by A = A0ekt. If k __________, the function models the amount, or size, of a growing entity. If k __________, the function models the amount, or size, of a decaying entity.
A. > 0; < 0
B. = 0; ≠ 0
C. ≥ 0; < 0
D. < 0; ≤ 0
12. Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
13. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
14. Find the domain of following logarithmic function.
f(x) = ln (x - 2)2
A. (∞, 2) ∪ (-2, -∞)
B. (-∞, 2) ∪ (2, ∞)
C. (-∞, 1) ∪ (3, ∞)
D. (2, -∞) ∪ (2, ∞)
15. Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
16. Find the domain of following logarithmic function.
f(x) = log5 (x + 4)
A. (-4, ∞)
B. (-5, -∞)
C. (7, -∞)
D. (-9, ∞)
17. Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.
ex = 5.7
18. Approximate the following using a calculator; round your answer to three decimal places.
e-0.95
A. .483
B. 1.287
C. .597
D. .387
19. You have $10,000 to invest. One bank pays 5% interest compounded quarterly and a second bank pays 4.5% interest compounded monthly. Use the formula for compound interest to write a function for the balance in each bank at any time t.
20. Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.
32x + 3x - 2 = 0
A. {1}
B. {-2}
C. {5}
D. {0}
21. Solve each equation by the substitution method.
x + y = 1
x2 + xy – y2 = -5
22. Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.
(-1, 6), (1, 4), (2, 9)
23. Solve the following system.
x + y + z = 6
3x + 4y - 7z = 1
2x - y + 3z = 5
24. On your next vacation, you will divide lodging between large resorts and small inns. Let x represent the number of nights spent in large resorts. Let y represent the number of nights spent in small inns.
Write a system of inequalities that models the following conditions:
You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average $200 per night and small inns average $100 per night. Your budget permits no more than $700 for lodging.
•
A. y ≥ 1
x + y ≥ 5
x ≥ 1
300x + 200y ≤ 700
B. y ≥ 0
x + y ≥ 3
x ≥ 0
200x + 200y ≤ 700
C. y ≥ 1
x + y ≥ 4
x ≥ 2
500x + 100y ≤ 700
D. y ≥ 0
x + y ≥ 5
x ≥ 1
200x + 100y ≤ 700
25. Solve each equation by either substitution or addition method.
x2 + 4y2 = 20
x + 2y = 6
26. Solve the following system by the addition method.
{4x + 3y = 15
{2x – 5y = 1
27. Let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers.
The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.
28. Solve the following system.
x = y + 4
3x + 7y = -18
29. Write the partial fraction decomposition for the following rational expression.
6x - 11/(x - 1)2
30. Perform the long division and write the partial fraction decomposition of the remainder term.
x4 – x2 + 2/x3 - x2
31. Write the partial fraction decomposition for the following rational expression.
4/2x2 - 5x – 3
32. Write the form of the partial fraction decomposition of the rational expression.
7x - 4/x2 - x - 12
33. Write the partial fraction decomposition for the following rational expression.
x + 4/x2(x + 4)
34. Perform the long division and write the partial fraction decomposition of the remainder term.
x5 + 2/x2 - 1
35. Solve the following system.
2x + 4y + 3z = 2
x + 2y - z = 0
4x + y - z = 6
36. Write the form of the partial fraction decomposition of the rational expression.
5x2 - 6x + 7/(x - 1)(x2 + 1)
37. Solve each equation by the substitution method.
x2 - 4y2 = -7
3x2 + y2 = 31
38. Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.
(-1, -4), (1, -2), (2, 5)
39. Write the partial fraction decomposition for the following rational expression.
x2 – 6x + 3/(x – 2)3
40. Solve the following system.
3(2x+y) + 5z = -1
2(x - 3y + 4z) = -9
4(1 + x) = -3(z - 3y)
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