AP PreCalculus Practice Quiz: Logarithmic Expressions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

Evaluate the logarithmic expression log₂(8).

A)3B)4C)16D)64

All Questions (10)

Evaluate the logarithmic expression log₂(8).

A) 3

B) 4

C) 16

D) 64

Correct Answer: A

The expression log₂(8) asks for the value that the base 2 must be exponentially raised to in order to obtain 8. Since 2³ = 8, the value of the expression is 3.

Which of the following best defines the expression logₐ(x)?

A) The value of a multiplied by x.

B) The value that x must be raised to in order to obtain a.

C) The value that the base a must be exponentially raised to in order to obtain the value x.

D) The value of a divided by x.

Correct Answer: C

This is the fundamental definition of a logarithm. The logarithmic expression logₐ(x) is equal to the exponent that the base 'a' must be raised to in order to get the value 'x'.

What is the value of the expression log₅(1/25)?

A) 2

B) -2

C) 1/2

D) -5

Correct Answer: B

The expression asks for the power to which 5 must be raised to get 1/25. Since 5² = 25, it follows that 5⁻² = 1/5² = 1/25. Therefore, the value is -2.

Which of the following logarithmic expressions can be readily evaluated through basic arithmetic without the use of technology?

A) log₃(12)

B) log₅(20)

C) log₁₀(1000)

D) log₂(9)

Correct Answer: C

The expression log₁₀(1000) can be evaluated using basic arithmetic because 1000 is a well-known integer power of the base 10 (10³ = 1000). The other options involve values that are not integer powers of their respective bases and would require technology to estimate.

On a logarithmic scale with a base of 10, an increase of one unit on the scale corresponds to what change in the measured quantity?

A) An additive change of 10.

B) A multiplicative change of 10.

C) An additive change of 1.

D) A multiplicative change of 1.

Correct Answer: B

On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. Since the base is 10, an increase of one unit means the actual quantity is multiplied by 10.

Evaluate the expression log₉(3).

A) 1/3

B) 3

C) 1/2

D) 2

Correct Answer: C

The expression log₉(3) asks for the power to which the base 9 must be raised to obtain 3. Since the square root of 9 is 3, and a square root is represented by the exponent 1/2, we have 9¹ᐟ² = 3. Thus, the value is 1/2.

Which of the following expressions would most likely require the use of technology to estimate its value?

A) log₄(64)

B) log₇(50)

C) log₃(81)

D) log₂(1/16)

Correct Answer: B

The value of log₇(50) cannot be determined through simple arithmetic, as 50 is not an integer power of 7. The other options can be evaluated as log₄(64) = 3, log₃(81) = 4, and log₂(1/16) = -4.

On a logarithmic scale with base 'b', a measurement increases from 4 to 6 on the scale. By what factor has the actual quantity increased?

A) 2

B) 2b

C) b²

D) b⁶ - b⁴

Correct Answer: C

Each unit increase on a logarithmic scale represents a multiplicative change of the base, 'b'. An increase from 4 to 6 is an increase of 2 units. Therefore, the actual quantity has been multiplied by the base twice, resulting in a total multiplicative factor of b × b = b².

What is the value of logₓ(1) for any valid base x (where x > 0 and x ≠ 1)?

A) x

B) 1

C) -1

D) 0

Correct Answer: D

The expression logₓ(1) asks for the exponent to which the base x must be raised to obtain 1. Any valid base raised to the power of 0 is equal to 1 (x⁰ = 1). Therefore, the value of the expression is 0.

Evaluate the logarithmic expression log₆₄(4).

A) 3

B) 16

C) 1/3

D) 1/4

Correct Answer: C

The expression log₆₄(4) asks for the value that the base 64 must be exponentially raised to in order to obtain 4. Since the cube root of 64 is 4 (4³ = 64), and a cube root can be written as an exponent of 1/3, we have 64¹ᐟ³ = 4. Therefore, the value is 1/3.