Algebra 2 Practice Test: Exponential and Logarithmic Functions
- Exponential and Logarithmic Functions
- 15 questions
- 20 minutes
These 15 questions practice evaluating logarithms, converting between exponential and logarithmic form, log properties, exponential and logarithmic equations, growth and decay models, half-life, compound interest and inverse functions. logb x = y means by = x.
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Question 1
What is the value of log2 32?
Correct answer: A
log2 32 asks which power of 2 equals 32. Since 25 = 32, the value is 5. The answer 16 comes from dividing 32 by 2.
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Question 2
Which equation is equivalent to log3 81 = 4?
Correct answer: C
logb x = y means by = x. Here the base is 3, the exponent is 4 and the result is 81, so 34 = 81.
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Question 3
Solve 2x + 3 = 64.
Correct answer: A
64 = 26, so x + 3 = 6 and x = 3. Writing x + 3 = 32 (half of 64) gives 29.
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Question 4
Solve 4x = 8x − 2.
Correct answer: D
Write both sides with base 2: 22x = 23x − 6. Then 2x = 3x − 6, so x = 6. Check: 46 = 4096 and 84 = 4096.
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Question 5
Which expression is equivalent to log(x3y) for positive x and y?
Correct answer: B
The quotient rule turns division into subtraction, and the power rule moves the exponent in front: log x3 − log y = 3log x − log y. There is no rule that turns a quotient into log xlog y.
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Question 6
What is the value of 2log5 10 − log5 4?
Correct answer: C
2log5 10 = log5 100, and log5 100 − log5 4 = log5 25 = 2. Subtracting before applying the power rule gives log5 16, which is not equal.
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Question 7
Solve log3(2x − 1) = 2.
Correct answer: D
Rewrite as 32 = 2x − 1, so 2x − 1 = 9 and x = 5. Using 3 · 2 = 6 instead of 32 gives 3.5.
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Question 8
Solve log2 x + log2(x − 2) = 3.
Correct answer: B
log2(x(x − 2)) = 3, so x2 − 2x = 8 and (x − 4)(x + 2) = 0. The value −2 is extraneous because the logarithm of a negative number is undefined, so only x = 4 works.
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Question 9
A town has 2,400 residents, and its population grows by 3% per year. Which function gives the population after t years?
Correct answer: A
Each year the population is multiplied by 1 + 0.03 = 1.03. A factor of 0.03 would shrink the population, and 1.3 represents 30% growth.
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Question 10
A car bought for $28,000 loses 15% of its value each year. What is its value after 3 years?
Correct answer: B
Multiply by 0.85 each year: 28000(0.85)3 = 17195.50. Subtracting 15% of the original price three times gives $15,400, and $42,584.50 uses growth (1.15)3 instead of decay.
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Question 11
A medicine has a half-life of 6 hours. A patient has 80 mg in the body. How many milligrams remain after 24 hours?
Correct answer: A
24 hours is 246 = 4 half-lives, so 80(12)4 = 80 · 116 = 5 mg. The answer 10 mg is what remains after only 3 half-lives (18 hours).
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Question 12
What is the solution of 5x = 40, rounded to the nearest hundredth?
Correct answer: C
Take logarithms: x = log 40log 5 ≈ 1.6020.699 ≈ 2.29. Check: 52 = 25 and 53 = 125, so x must be between 2 and 3. The value 0.44 is the fraction inverted.
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Question 13
What is the horizontal asymptote of the graph of f(x) = 3x − 4?
Correct answer: D
The graph of y = 3x approaches y = 0 as x → −∞. Subtracting 4 shifts the graph down 4 units, so the asymptote becomes y = −4.
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Question 14
What is the inverse of f(x) = ln(x − 1)?
Correct answer: B
Swap x and y: x = ln(y − 1), so ex = y − 1 and y = ex + 1. Check: f(ex + 1) = ln(ex) = x. The inverse function is not the reciprocal 1f(x).
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Question 15
Ana invests $5,000 at an annual interest rate of 4%, compounded quarterly. Which expression gives the value of the investment after 10 years?
Correct answer: C
Use A = P(1 + rn)nt with r = 0.04, n = 4 and t = 10: the rate per quarter is 0.01 and there are 40 quarters. The result is about $7,444.32.
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