2.10 Inverses of Exponential Functions

Exponential functions and logarithmic functions are inverses of each other: if $g(x) = b^x$, then $f(x) = \log_b x$ undoes it, so $g(f(x)) = f(g(x)) = x$. Their graphs are reflections across the line $y = x$, which means every point $(s, t)$ on $b^x$ becomes the point $(t, s)$ on $\log_b x$.

Why This Matters for the AP Precalculus Exam

This topic ties together two function families you will use constantly in Unit 2, which carries a large share of the exam. Recognizing that $\log_b x$ is the inverse of $b^x$ lets you switch between exponential and logarithmic forms, read graphs as reflections across $y = x$, and swap input-output pairs without redoing all the work. You will see these ideas in both multiple-choice and free-response questions, often asking you to move between graphical, numerical, and analytical representations of the same relationship. Clear notation and steps are important for clear exam work, especially when you show why two functions are inverses.

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Key Takeaways

• The general form of a logarithmic function is $f(x) = a\log_b x$, where $b > 0$, $b \neq 1$, and $a \neq 0$.
• $f(x) = \log_b x$ and $g(x) = b^x$ are inverse functions, so $g(f(x)) = f(g(x)) = x$.
• The graph of $\log_b x$ is the reflection of the graph of $b^x$ across the identity line $h(x) = x$.
• If $(s, t)$ is a point on $g(x) = b^x$, then $(t, s)$ is a point on $f(x) = \log_b x$.
• Exponential growth means output changes multiplicatively as input changes additively; logarithmic growth means output changes additively as input changes multiplicatively.
• Domain and range swap: $b^x$ has range $(0, \infty)$, and $\log_b x$ has domain $(0, \infty)$.

Connecting Exponential and Logarithmic Functions

This builds directly on inverse functions from Topic 2.8 and logarithmic expressions from Topic 2.9. A logarithmic function is defined as $f(x) = a\log_b x$, where $b$ is the base of the logarithm and $a$ is the coefficient. The base $b$ must be greater than 0 and not equal to 1, and the coefficient $a$ must not be 0.

You can see the inverse relationship between input and output values by comparing the general forms. The general form of an exponential function is $f(x) = ab^x$, where $a$ is the coefficient and $b$ is the base. As the input value $x$ increases in equal-length intervals, the output value $f(x)$ increases proportionately.

For a logarithmic function $f(x) = a\log_b x$, the relationship flips. As the output value $f(x)$ increases in equal-length intervals, the input value $x$ increases proportionately. That flip is the whole point of an inverse.

How Inverse Growth Works

Exponential growth is characterized by output values changing multiplicatively as input values change additively. Logarithmic growth is the reverse: output values change additively as input values change multiplicatively. This happens because in exponential functions the input value is the exponent, while in logarithmic functions the input value is the argument of the logarithm.

Reflections and the Identity Function

The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, across the graph of the identity function $h(x) = x$. This comes straight from the inverse relationship between the two functions.

The exponential function $g(x) = b^x$ has a curve that rises quickly as $x$ increases when $b > 1$. Its graph has a horizontal asymptote at $y = 0$ and no vertical asymptote.

The logarithmic function $f(x) = \log_b x$ has a curve that rises slowly as $x$ increases when $b > 1$. Its graph has a vertical asymptote at $x = 0$ and no horizontal asymptote.

The identity function $h(x) = x$ is a straight line with slope 1 and y-intercept 0, and it has no asymptotes. When you reflect the graph of $g(x)$ across $h(x)$, you get the graph of $f(x)$, with the x- and y-coordinates switched. That switch is exactly what an inverse does to every point.

Applying the Relationship: Ordered Pairs

If $(s, t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \neq 1$, then $(t, s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

The exponential function $g(x) = b^x$ takes input $s$ and raises the base $b$ to that power, giving output $t$. So $t = b^s$.

The logarithmic function $f(x) = \log_b x$ takes input $t$ and finds the exponent the base $b$ must be raised to in order to equal $t$, giving output $s$. So $s = \log_b t$.

Because the two functions are inverses, the same numbers $s$ and $t$ describe both equations. The statements $t = b^s$ and $s = \log_b t$ are just two ways of writing the same fact, which is why you can swap an ordered pair from one function to get a point on the other.

How to Use This on the AP Precalculus Exam

MCQ

• When a question gives you a point on $b^x$, swap the coordinates to get the matching point on $\log_b x$, and vice versa.
• If you see a graph reflected across $y = x$, expect an exponential and logarithmic pair with the same base.
• Use $t = b^s$ if and only if $s = \log_b t$ to rewrite an equation in the form that is easier to evaluate.

Free Response

• To show two functions are inverses, demonstrate that $g(f(x)) = f(g(x)) = x$.
• When building or sketching a graph, label the asymptotes: $y = 0$ for $b^x$ and $x = 0$ for $\log_b x$.
• Show each step when you convert between exponential and logarithmic form so your reasoning is clear.

Common Trap

• The domain and range trade places between the two functions, so do not assume a logarithm accepts negative or zero inputs.

Common Misconceptions

• The asymptotes are not the same for both functions. The exponential $b^x$ has a horizontal asymptote at $y = 0$, while the logarithm $\log_b x$ has a vertical asymptote at $x = 0$.
• Reflecting across $y = x$ is not the same as reflecting across the x-axis or y-axis. Only reflection across the identity line produces the inverse.
• An ordered pair swap gives $(t, s)$ from $(s, t)$. Students sometimes only swap the sign or change one coordinate, which breaks the inverse relationship.
• $\log_b x$ is only defined for $x > 0$. The output of $b^x$ is always positive, so its inverse cannot take in zero or negative numbers.
• A logarithmic function is not the reciprocal of an exponential function. Inverse means undoing the mapping, not dividing 1 by the function.

Related AP Precalculus Guides

• 2.8 Inverse Functions
• 2.11 Logarithmic Functions
• 2.4 Exponential Function Manipulation
• 2.7 Composition of Functions
• 2.12 Logarithmic Function Manipulation
• 2.3 Exponential Functions