AP PreCalculus Flashcards: Inverses of Exponential Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 14 cards to help you master important concepts.
If the point (s, t) is on the graph of the exponential function g(x) = b^{x}, what corresponding point must be on the graph of its inverse, f(x) = log_{b}x?
If (s, t) is an ordered pair of the exponential function, then (t, s) is an ordered pair of its inverse logarithmic function.
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If the point (s, t) is on the graph of the exponential function g(x) = b^{x}, what corresponding point must be on the graph of its inverse, f(x) = log_{b}x?
If (s, t) is an ordered pair of the exponential function, then (t, s) is an ordered pair of its inverse logarithmic function.
If (5, 32) is a point on the graph of g(x) = 2^{x}, what is the corresponding point on the graph of f(x) = log_{2}x?
The corresponding point on the graph of the inverse logarithmic function f(x) = log_{2}x is (32, 5).
A function's output increases by 1 every time the input is multiplied by 10. What kind of function is this?
This represents logarithmic growth, as the output changes additively while the input changes multiplicatively.
What conditions must be met for the base, b, in a logarithmic function?
The base b must be a positive number and cannot be equal to 1 (b > 0 and b ≠ 1).
What is the line of reflection between the graph of an exponential function and its inverse logarithmic function?
The line of reflection is the graph of the identity function, h(x) = x.
How can you represent the inverse of an exponential function with an initial value of 1?
The inverse can be represented graphically as a reflection over y=x, or algebraically as a logarithmic function f(x) = log_{b}x.
What is the general form of a logarithmic function?
The general form is f(x) = a log_{b}x, where the base b > 0, b ≠ 1, and a ≠ 0.
Explain the relationship between the ordered pairs of f(x) = log_{b}x and g(x) = b^{x}.
For every ordered pair (s, t) on the exponential function, there is a corresponding ordered pair (t, s) on the logarithmic function.
Define logarithmic growth.
Logarithmic growth is characterized by output values changing additively as input values change multiplicatively.
A function exhibits logarithmic growth. If its input is multiplied by a constant factor, how will its output change?
The output will change by a constant additive amount, as output values change additively when input values change multiplicatively.
What type of function is the inverse of an exponential function with an initial value of 1?
The inverse of an exponential function with an initial value of 1 (e.g., y = b^x) is a logarithmic function.
How are the graphs of an exponential function g(x) = b^{x} and its inverse logarithmic function f(x) = log_{b}x related?
The graph of the logarithmic function f(x) = log_{b}x is a reflection of the graph of the exponential function g(x) = b^{x} over the line y = x.
What is the functional relationship between f(x) = log_{b}x and g(x) = b^{x}?
The functions f(x) = log_{b}x and g(x) = b^{x} (where b > 0 and b ≠ 1) are inverse functions of each other.
Inverse of g(x) = b^{x}
The inverse of the exponential function g(x) = b^{x} is the logarithmic function f(x) = log_{b}x.