Topic 2.10 Inverses of Exponential Functions: construct the inverse of an exponential function as a logarithmic function, and relate the graph, domain and range of each to the other.

What this topic is asking

The College Board (Topic 2.10) wants you to see the logarithm as the inverse of the exponential function. You must construct $f^{-1}$ for $f(x) = b^x$ (it is $\log_b x$), and relate the graph, domain and range of the exponential to those of its logarithmic inverse, including the reflection over the line $y = x$.

The logarithm undoes the exponential

Finding the inverse uses the swap-and-solve method from Topic 2.8: write $y = b^x$, swap to $x = b^y$, and "solve for $y$" by taking a base-$b$ logarithm, giving $y = \log_b x$.

Graphs reflect over y = x

Because they are inverses, the graphs of $b^x$ and $\log_b x$ are mirror images across the line $y = x$. Every point $(p, q)$ on the exponential corresponds to $(q, p)$ on the logarithm. The exponential rises steeply and flattens toward its horizontal asymptote; the logarithm, reflected, rises slowly and falls steeply toward its vertical asymptote.

Why the asymptotes swap roles

The reflection over $y = x$ turns horizontal features into vertical ones. The exponential approaches the horizontal line $y = 0$ as $x \to -\infty$; reflecting that line over $y = x$ gives the vertical line $x = 0$, which is the logarithm's vertical asymptote as $x \to 0^+$. This is why a logarithm "blows down" to $-\infty$ near $x = 0$ rather than near a horizontal line. Tracing the asymptote through the reflection, rather than memorizing it separately, ties the two graphs together and explains the logarithm's shape directly from the exponential's.

A clarifying point is that the inverse of an exponential is a logarithm, not a reciprocal or a reflection like $b^{-x}$. The function $2^{-x}$ is a decay curve (a horizontal reflection), and $\frac{1}{2^x}$ is the same thing, but neither undoes $2^x$. Only $\log_2 x$ returns the original input, which is the defining property of an inverse and the reason logarithms exist as a tool for solving exponential equations in Topic 2.13.

Try this

Q1. What is the inverse of $f(x) = 10^{x}$? [1 point]

• Cue. $f^{-1}(x) = \log_{10} x = \log x$, the common logarithm.

Q2. The exponential $b^x$ has range $y > 0$. What is the domain of its inverse $\log_b x$? [1 point]

• Cue. $x > 0$: the range of the function becomes the domain of the inverse.