Precalculus Practice Test: Logarithms
- Logarithms
- 10 questions
- 15 minutes
These 10 questions cover evaluating logarithms, the product, quotient and power rules, change of base and logarithmic equations. After solving a log equation, always check your answers: the argument of every logarithm must be positive.
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Question 1
What is the value of log2 32?
Correct answer: D
Since 25 = 32, log2 32 = 5. The value 16 comes from dividing 32 by 2 instead of asking which power of 2 equals 32.
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Question 2
If log3 x = 4, what is x?
Correct answer: B
In exponential form the equation says 34 = x, so x = 81. The value 64 = 43 swaps the base and the exponent.
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Question 3
Which expression is equivalent to log(x3yz)?
Correct answer: A
The product rule splits x3y, the quotient rule subtracts log z, and the power rule turns log x3 into 3 log x. The exponent 3 belongs only to x, so it must not multiply log y.
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Question 4
What is the solution of log2(x + 3) = 4?
Correct answer: C
Rewrite as x + 3 = 24 = 16, so x = 13. The value 16 forgets to subtract 3, and 5 comes from computing 2·4 − 3, using 2·4 instead of 24.
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Question 5
What is the solution of 5x = 20, rounded to three decimal places?
Correct answer: D
Take logarithms: x = ln 20 ÷ ln 5 ≈ 2.9957 ÷ 1.6094 ≈ 1.861. The value 0.537 divides the logarithms in the wrong order, and 1.301 is just log 20.
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Question 6
What is the value of log4 2 + log4 8?
Correct answer: B
By the product rule, log4 2 + log4 8 = log4 16 = 2, because 42 = 16. Writing log4 10 adds the arguments, but the rule multiplies them.
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Question 7
What is the domain of f(x) = ln(5 − x)?
Correct answer: A
The argument must be positive: 5 − x > 0, so x < 5. The interval (−∞, 5] includes 5, where the argument is 0 and the logarithm is undefined.
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Question 8
What are all the solutions of log x + log(x − 3) = 1? (log means base 10.)
Correct answer: C
Combine: log(x(x − 3)) = 1, so x2 − 3x − 10 = 0 and x = 5 or x = −2. The value −2 must be rejected because log(−2) is undefined.
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Question 9
The pH of a solution is pH = −log[H+]. How many times greater is the hydrogen ion concentration of a solution with pH 3 than of a solution with pH 5?
Correct answer: B
A difference of 2 pH units means a factor of 102 = 100 in [H+], and the lower pH has the higher concentration. The answer 2 treats the logarithmic scale as if it were linear.
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Question 10
What is the exact value of log8 32?
Correct answer: D
Write both numbers as powers of 2: log8 32 = log2 32 ÷ log2 8 = 5/3. Check: 85/3 = (23)5/3 = 25 = 32. The fraction 3/5 is the change-of-base ratio upside down.
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