College Algebra Help
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college_algebra_1.docx1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
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A. 1/2 logb x - 6 logb y + 3 logb z |
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B. 1/2 logb x - 9 logb y - 3 logb z |
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C. 1/2 logb x + 3 logb y + 6 logb z |
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D. 1/2 logb x + 3 logb y - 3 logb 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
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
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
5) 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.
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
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?
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
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 __________.
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
11) Write the following equation in its equivalent exponential form.
4 = log2 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
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.
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.
ex = 5.7
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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.
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A. x; y = 0; horizontal |
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B. x; y = 1; vertical |
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C. -x; y = 0; horizontal |
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D. x; y = -1; vertical |
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17) 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.
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A. > 0; < 0 |
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B. = 0; ≠ 0 |
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C. ≥ 0; < 0 |
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D. < 0; ≤ 0 |
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18) Find the domain of following logarithmic function.
f(x) = ln (x - 2)2
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A. (∞, 2) ∪ (-2, -∞) |
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B. (-∞, 2) ∪ (2, ∞) |
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C. (-∞, 1) ∪ (3, ∞) |
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D. (2, -∞) ∪ (2, ∞) |
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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?
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A. 10 grams after 10 seconds; 6 grams after 20 seconds |
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B. 12 grams after 10 seconds; 7 grams after 20 seconds |
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C. 4 grams after 10 seconds; 1 gram after 20 seconds |
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D. 8 grams after 10 seconds; 4 grams after 20 seconds |
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20) Evaluate the following expression without using a calculator.
Log7 √7
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A. 1/4 |
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B. 3/5 |
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C. 1/2 |
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D. 2/7 |
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21) Write the form of the partial fraction decomposition of the rational expression.
7x - 4/x2 - x - 12
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A. 24/7(x - 2) + 26/7(x + 5) |
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B. 14/7(x - 3) + 20/7(x2 + 3) |
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C. 24/7(x - 4) + 25/7(x + 3) |
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D. 22/8(x - 2) + 25/6(x + 4) |
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22) Solve the following system.
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2x + 4y + 3z = 2 x + 2y - z = 0 4x + y - z = 6 |
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A. {(-3, 2, 6)} |
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B. {(4, 8, -3)} |
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C. {(3, 1, 5)} |
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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.
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A. x + y = 7; x - y = -1; 3 and 4 |
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B. x + y = 7; x - y = -1; 5 and 6 |
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C. x + y = 7; x - y = -1; 3 and 6 |
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D. x + y = 7; x - y = -1; 2 and 3 |
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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.
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A. 50x + 150y > 2000 |
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B. 100x + 150y > 1000 |
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C. 70x + 250y > 2000 |
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D. 55x + 150y > 3000 |
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25) Solve the following system.
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3(2x+y) + 5z = -1 2(x - 3y + 4z) = -9 4(1 + x) = -3(z - 3y) |
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A. {(1, 1/3, 0)} |
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B. {(1/4, 1/3, -2)} |
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C. {(1/3, 1/5, -1)} |
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D. {(1/2, 1/3, -1)} |
26) Solve the following system.
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x + y + z = 6 3x + 4y - 7z = 1 2x - y + 3z = 5 |
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A. {(1, 3, 2)} |
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B. {(1, 4, 5)} |
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C. {(1, 2, 1)} |
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D. {(1, 5, 7)} |
27) Solve the following system.
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2x + y = 2 x + y - z = 4 3x + 2y + z = 0 |
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A. {(2, 1, 4)} |
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B. {(1, 0, -3)} |
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C. {(0, 0, -2)} |
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D. {(3, 2, -1)} |
28) Write the form of the partial fraction decomposition of the rational expression.
5x2 - 6x + 7/(x - 1)(x2 + 1)
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A. A/x - 2 + Bx2 + C/x2 + 3 |
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B. A/x - 4 + Bx + C/x2 + 1 |
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C. A/x - 3 + Bx + C/x2 + 1 |
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D. A/x - 1 + Bx + C/x2 + 1 |
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29) Solve the following system by the addition method.
{4x + 3y = 15
{2x – 5y = 1
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A. {(4, 0)} |
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B. {(2, 1)} |
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C. {(6, 1)} |
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D. {(3, 1)} |
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30) Write the partial fraction decomposition for the following rational expression.
6x - 11/(x - 1)2
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A. 6/x - 1 - 5/(x - 1)2 |
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B. 5/x - 1 - 4/(x - 1)2 |
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C. 2/x - 1 - 7/(x - 1) |
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D. 4/x - 1 - 3/(x - 1) |
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31) Write the partial fraction decomposition for the following rational expression.
x2 – 6x + 3/(x – 2)3
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A. 1/x – 4 – 2/(x – 2)2 – 6/(x – 2) |
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B. 1/x – 2 – 4/(x – 2)2 – 5/(x – 1)3 |
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C. 1/x – 3 – 2/(x – 3)2 – 5/(x – 2) |
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D. 1/x – 2 – 2/(x – 2)2 – 5/(x – 2)3 |
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32) Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.
(-1, 6), (1, 4), (2, 9)
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A. y = 2x2 - x + 3 |
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B. y = 2x2 + x2 + 9 |
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C. y = 3x2 - x - 4 |
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D. y = 2x2 + 2x + 4 |
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33) Perform the long division and write the partial fraction decomposition of the remainder term.
x5 + 2/x2 - 1
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A. x2 + x - 1/2(x + 1) + 4/2(x - 1) |
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B. x3 + x - 1/2(x + 1) + 3/2(x - 1) |
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C. x3 + x - 1/6(x - 2) + 3/2(x + 1) |
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D. x2 + x - 1/2(x + 1) + 4/2(x - 1) |
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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.
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A. z = 200x + 125y |
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B. z = 125x + 200y |
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C. z = 130x + 225y |
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D. z = -125x + 200y |
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35) Solve each equation by the substitution method.
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y2 = x2 - 9 2y = x – 3 |
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A. {(-6, -4), (2, 0)} |
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B. {(-4, -4), (1, 0)} |
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C. {(-3, -4), (2, 0)} |
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D. {(-5, -4), (3, 0)} |
26) Solve each equation by the substitution method.
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x + y = 1 x2 + xy – y2 = -5 |
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A. {(4, -3), (-1, 2)} |
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B. {(2, -3), (-1, 6)} |
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C. {(-4, -3), (-1, 3)} |
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D. {(2, -3), (-1, -2)} |
37) Solve each equation by the addition method.
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x2 + y2 = 25 (x - 8)2 + y2 = 41 |
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A. {(3, 5), (3, -2)} |
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B. {(3, 4), (3, -4)} |
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C. {(2, 4), (1, -4)} |
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D. {(3, 6), (3, -7)} |
38) Write the partial fraction decomposition for the following rational expression.
ax +b/(x – c)2 (c ≠ 0)
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A. a/a – c +ac + b/(x – c)2 |
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B. a/b – c +ac + b/(x – c) |
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C. a/a – b +ac + c/(x – c)2 |
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D. a/a – b +ac + b/(x – c) |
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39) Solve each equation by either substitution or addition method.
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x2 + 4y2 = 20 x + 2y = 6 |
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A. {(5, 2), (-4, 1)} |
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B. {(4, 2), (3, 1)} |
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C. {(2, 2), (4, 1)} |
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D. {(6, 2), (7, 1)} |
