AP PreCalculus Practice Quiz: Inverses of Exponential Functions

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

Test your understanding with short quizzes. This quiz has 12 questions to check your progress.

Question 1 of 12

According to the definition of inverse functions, what is the inverse of the exponential function g(x) = b^{x}, where b > 0 and b ≠ 1?

A)f(x) = x^{b}B)f(x) = log_{b}xC)f(x) = -b^{x}D)f(x) = b^{-x}

All Questions (12)

According to the definition of inverse functions, what is the inverse of the exponential function g(x) = b^{x}, where b > 0 and b ≠ 1?

A) f(x) = x^{b}

B) f(x) = log_{b}x

C) f(x) = -b^{x}

D) f(x) = b^{-x}

Correct Answer: B

Content point 4 explicitly states that f(x) = log_{b}x and g(x) = b^{x} are inverse functions.

The graph of the logarithmic function f(x) = log_{b}x can be obtained by reflecting the graph of the exponential function g(x) = b^{x} over which line?

A) The x-axis

B) The y-axis

C) The line y = -x

D) The line y = x

Correct Answer: D

Content point 5 states that the graph of a logarithmic function is a reflection of its inverse exponential function over the graph of the identity function, h(x) = x.

If the point (3, 27) is on the graph of the exponential function g(x) = 3^{x}, which ordered pair must be on the graph of its inverse, f(x) = log_{3}x?

A) (3, 27)

B) (-3, 1/27)

C) (27, 3)

D) (1/3, 27)

Correct Answer: C

Based on content point 6, if (s, t) is an ordered pair of an exponential function, then (t, s) is an ordered pair of its inverse logarithmic function. Therefore, (3, 27) on the exponential function corresponds to (27, 3) on the logarithmic function.

Which of the following statements accurately describes the characteristic of logarithmic growth?

A) Output values change additively as input values also change additively.

B) Output values change multiplicatively as input values change additively.

C) Output values change multiplicatively as input values also change multiplicatively.

D) Output values change additively as input values change multiplicatively.

Correct Answer: D

This is the definition provided in content point 3: Logarithmic growth is characterized by output values changing additively as input values change multiplicatively.

For the general form of a logarithmic function, f(x) = a log_{b}x, which of the following are the required constraints for the base b?

A) b > 0 and b ≠ 1

B) b < 0

C) b = 1

D) b can be any real number except 0

Correct Answer: A

Content point 2 specifies the general form of a logarithmic function and states that the base b must satisfy the conditions b > 0 and b ≠ 1.

To construct a representation of the inverse of the exponential function y = 5^{x}, what is the resulting function?

A) y = log_{x}5

B) x = log_{y}5

C) y = log_{5}x

D) y = x^{5}

Correct Answer: C

Following content point 1, we construct the inverse. We start with y = 5^{x}. To find the inverse, we swap x and y to get x = 5^{y}. Then, we solve for y by converting to logarithmic form, which is y = log_{5}x. This matches the rule from content point 4.

The graph of an exponential function g(x) = b^{x} contains the point (m, n). Which of the following must be true about the graph of its inverse, f(x) = log_{b}x?

A) It is a reflection of g(x) over the y-axis and contains the point (-m, n).

B) It is a reflection of g(x) over the line y = x and contains the point (n, m).

C) It is a reflection of g(x) over the x-axis and contains the point (m, -n).

D) It is a reflection of g(x) over the line y = x and contains the point (m, n).

Correct Answer: B

This question combines two concepts. Content point 5 states that the graphs are reflections over y = x. Content point 6 states that if (s, t) is on the exponential function, (t, s) is on the logarithmic function. Therefore, the inverse graph is a reflection over y = x and contains the point (n, m).

A function f(x) exhibits logarithmic growth. Given that f(3) = 5 and f(9) = 7, what is the value of f(27)?

A) 8

B) 9

C) 11

D) 14

Correct Answer: B

According to content point 3, for logarithmic growth, output values change additively as input values change multiplicatively. The input changes from 3 to 9 by a factor of 3 (9/3 = 3), and the output changes from 5 to 7 by adding 2 (7-5 = 2). To get the next value, we multiply the input by 3 again (9 * 3 = 27), so we must add 2 to the output: 7 + 2 = 9.

The ordered pair (4, 16) is a solution to the exponential function g(x) = 2^{x}. What is the corresponding ordered pair for the inverse logarithmic function f(x) = log_{2}x, and what geometric relationship does this demonstrate?

A) (16, 4), demonstrating a reflection over the line y = x.

B) (4, 16), demonstrating that the functions are identical.

C) (-4, 1/16), demonstrating a reflection over the y-axis.

D) (16, 4), demonstrating a reflection over the x-axis.

Correct Answer: A

This question synthesizes content points 5 and 6. From point 6, if (4, 16) is on g(x), then (16, 4) is on its inverse f(x). From point 5, this swapping of coordinates is the algebraic result of the geometric reflection of the graph over the line y = x.

An exponential function with an initial value of 1 is reflected over the graph of the identity function h(x) = x to produce a new graph. If the point (k, j) is on the original exponential graph, which point must be on the new graph?

A) (k, j)

B) (-k, j)

C) (k, -j)

D) (j, k)

Correct Answer: D

This question combines several concepts. An exponential function with an initial value of 1 is g(x) = b^x (Content 1). Its reflection over the identity function h(x)=x is its inverse, the logarithmic function f(x) = log_b(x) (Content 5). If (k, j) is an ordered pair of the exponential function, then (j, k) is the corresponding ordered pair on its inverse (Content 6).

A function's behavior is described as follows: every time the input value is multiplied by 8, the output value increases by 3. Based on the general form f(x) = a log_{b}x, which equation models this behavior?

A) f(x) = 8 log_{3}x

B) f(x) = 3 log_{8}x

C) f(x) = log_{3}(x) + 8

D) f(x) = log_{8}(x) + 3

Correct Answer: B

This is a direct application of content point 3. The input changes multiplicatively (by 8), and the output changes additively (by 3). This implies a logarithmic function with base 8, so it has the form f(x) = a log_{8}x. To find 'a', consider two points x_1 and x_2=8x_1. The change in output is f(x_2) - f(x_1) = a log_{8}(8x_1) - a log_{8}(x_1) = a(log_{8}(8) + log_{8}(x_1)) - a log_{8}(x_1) = a*log_{8}(8) = a*1 = a. Since the output increases by 3, a = 3. The function is f(x) = 3 log_{8}x.

The statement 5^3 = 125 corresponds to a point on the graph of g(x) = 5^x. Which statement corresponds to a point on the graph of the inverse function, f(x) = log_5(x)?

A) log_5(3) = 125

B) log_3(5) = 125

C) log_5(125) = 3

D) log_125(5) = 3

Correct Answer: C

The exponential statement 5^3 = 125 means the ordered pair (s, t) = (3, 125) is on the function g(x)=5^x. According to content point 6, the corresponding ordered pair on the inverse logarithmic function is (t, s) = (125, 3). This means f(125) = 3, which is written as log_5(125) = 3.