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1) Use properties of logarithms to expand the following logarithmic expression as much as possible.

Log_b (√xy³/ z³)

A. 1/2 log_bx - 6 log_by + 3 log_bz

B. 1/2 log_b x - 9 log_b y - 3 log_b z

C. 1/2 log_b x + 3 log_b y + 6 log_b z

D. 1/2 log_b x + 3 log_b y - 3 log_b z

2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.

2 log x = log 25

A. {12}
B. {5}
C. {-3}
D. {25}

3) 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

4) 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

5) Use the exponential growth model, A = A₀e^kt, 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 = A₀e^kt; ln = e^kt; ln 2 = ln e^kt; ln 2 = kt; ln 2/k = t
B. 2A₀ = A₀e; 2= e^kt; ln = ln e^kt; ln 2 = kt; ln 2/k = t
C. 2A₀ = A₀e^kt; 2= e^kt; ln 2 = ln e^kt; ln 2 = kt; ln 2/k = t
D. 2A₀ = A₀e^kt; 2 = e^kt; ln 1 = ln e^kt; ln 2 = kt; ln 2/k = t^oe

6) Find the domain of following logarithmic function.

f(x) = log (2 - x)

A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)

7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?

A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams

8) Use properties of logarithms to expand the following logarithmic expression as much as possible.

log_b (x² y) / z²

A. 2 log_b x + log_b y - 2 log_b z
B. 4 log_b x - log_b y - 2 log_b z
C. 2 log_b x + 2 log_b y + 2 log_b z
D. log_b x - log_b y + 2 log_b z

9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.

A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)

10) 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

11) Write the following equation in its equivalent exponential form.

4 = log² 16

A. 2 log₄ = 16
B. 2² = 4
C. 4⁴ = 256
D. 2⁴ = 16

12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

3^1-x = 1/27

A. {2}
B. {-7}
C. {4}
D. {3}

13) Use properties of logarithms to expand the following logarithmic expression as much as possible.

log_b (x²y)

A. 2 log_yx + log_xy
B. 2 log_bx + log_by
C. log_x - log_by
D. log_bx – log_xy

14) 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.

A. A = 20,000(1 + (0.06/4))^4t; A = 10,000(1 + (0.044/14))^12t
B. A = 15,000(1 + (0.07/4))^4t; A = 10,000(1 + (0.025/12))^12t
C. A = 10,000(1 + (0.05/4))^4t; A = 10,000(1 + (0.045/12))^12t
D. A = 25,000(1 + (0.05/4))^4t; A = 10,000(1 + (0.032/14))^12t

15) 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.

e^x = 5.7

A. {ln 5.7}; ≈1.74
B. {ln 8.7}; ≈3.74
C. {ln 6.9}; ≈2.49
D. {ln 8.9}; ≈3.97

16) 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

17) Consider the model for exponential growth or decay given by A = A₀e^kt. 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

18) 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, ∞)

19) 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

20) Evaluate the following expression without using a calculator.

Log₇ √7

A. 1/4
B. 3/5
C. 1/2
D. 2/7

21) Write the form of the partial fraction decomposition of the rational expression.

7x - 4/x² - x - 12

A. 24/7(x - 2) + 26/7(x + 5)
B. 14/7(x - 3) + 20/7(x² + 3)
C. 24/7(x - 4) + 25/7(x + 3)
D. 22/8(x - 2) + 25/6(x + 4)

22) Solve the following system.

2x + 4y + 3z = 2

x + 2y - z = 0

4x + y - z = 6

A. {(-3, 2, 6)}
B. {(4, 8, -3)}
C. {(3, 1, 5)}
D. {(1, 4, -1)}

23) 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.

A. x + y = 7; x - y = -1; 3 and 4
B. x + y = 7; x - y = -1; 5 and 6
C. x + y = 7; x - y = -1; 3 and 6
D. x + y = 7; x - y = -1; 2 and 3

24) Many elevators have a capacity of 2000 pounds.

If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when x children and y adults will cause the elevator to be overloaded.

A. 50x + 150y > 2000
B. 100x + 150y > 1000
C. 70x + 250y > 2000
D. 55x + 150y > 3000

25) Solve the following system.

3(2x+y) + 5z = -1

2(x - 3y + 4z) = -9

4(1 + x) = -3(z - 3y)

A. {(1, 1/3, 0)}
B. {(1/4, 1/3, -2)}
C. {(1/3, 1/5, -1)}
D. {(1/2, 1/3, -1)}

26) Solve the following system.

x + y + z = 6

3x + 4y - 7z = 1

2x - y + 3z = 5

A. {(1, 3, 2)}
B. {(1, 4, 5)}
C. {(1, 2, 1)}
D. {(1, 5, 7)}

27) Solve the following system.

2x + y = 2

x + y - z = 4

3x + 2y + z = 0

A. {(2, 1, 4)}
B. {(1, 0, -3)}
C. {(0, 0, -2)}
D. {(3, 2, -1)}

28) Write the form of the partial fraction decomposition of the rational expression.

5x² - 6x + 7/(x - 1)(x² + 1)

A. A/x - 2 + Bx² + C/x² + 3
B. A/x - 4 + Bx + C/x² + 1
C. A/x - 3 + Bx + C/x² + 1
D. A/x - 1 + Bx + C/x² + 1

29) Solve the following system by the addition method.

{4x + 3y = 15

{2x – 5y = 1

A. {(4, 0)}
B. {(2, 1)}
C. {(6, 1)}
D. {(3, 1)}

30) Write the partial fraction decomposition for the following rational expression.

6x - 11/(x - 1)²

A. 6/x - 1 - 5/(x - 1)²
B. 5/x - 1 - 4/(x - 1)²
C. 2/x - 1 - 7/(x - 1)
D. 4/x - 1 - 3/(x - 1)

31) Write the partial fraction decomposition for the following rational expression.

x² – 6x + 3/(x – 2)³

A. 1/x – 4 – 2/(x – 2)² – 6/(x – 2)
B. 1/x – 2 – 4/(x – 2)² – 5/(x – 1)³
C. 1/x – 3 – 2/(x – 3)² – 5/(x – 2)
D. 1/x – 2 – 2/(x – 2)² – 5/(x – 2)³

32) Find the quadratic function y = ax² + bx + c whose graph passes through the given points.

(-1, 6), (1, 4), (2, 9)

A. y = 2x² - x + 3
B. y = 2x² + x² + 9
C. y = 3x² - x - 4
D. y = 2x² + 2x + 4

33) Perform the long division and write the partial fraction decomposition of the remainder term.

x⁵ + 2/x² - 1

A. x² + x - 1/2(x + 1) + 4/2(x - 1)
B. x³ + x - 1/2(x + 1) + 3/2(x - 1)
C. x³ + x - 1/6(x - 2) + 3/2(x + 1)
D. x² + x - 1/2(x + 1) + 4/2(x - 1)

34) A television manufacturer makes rear-projection and plasma televisions. The proﬁt per unit is $125 for the rear-projection televisions and $200 for the plasma televisions.

Let x = the number of rear-projection televisions manufactured in a month and let y = the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit.

A. z = 200x + 125y
B. z = 125x + 200y
C. z = 130x + 225y
D. z = -125x + 200y

35) Solve each equation by the substitution method.

y² = x² - 9

2y = x – 3

A. {(-6, -4), (2, 0)}
B. {(-4, -4), (1, 0)}
C. {(-3, -4), (2, 0)}
D. {(-5, -4), (3, 0)}

26) Solve each equation by the substitution method.

x + y = 1

x² + xy – y² = -5

A. {(4, -3), (-1, 2)}
B. {(2, -3), (-1, 6)}
C. {(-4, -3), (-1, 3)}
D. {(2, -3), (-1, -2)}

37) Solve each equation by the addition method.

x² + y² = 25

(x - 8)² + y² = 41

A. {(3, 5), (3, -2)}
B. {(3, 4), (3, -4)}
C. {(2, 4), (1, -4)}
D. {(3, 6), (3, -7)}

38) Write the partial fraction decomposition for the following rational expression.

ax +b/(x – c)² (c ≠ 0)

A. a/a – c +ac + b/(x – c)²
B. a/b – c +ac + b/(x – c)
C. a/a – b +ac + c/(x – c)²
D. a/a – b +ac + b/(x – c)

39) Solve each equation by either substitution or addition method.

x² + 4y² = 20

x + 2y = 6

A. {(5, 2), (-4, 1)}
B. {(4, 2), (3, 1)}
C. {(2, 2), (4, 1)}
D. {(6, 2), (7, 1)}

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