Inverses of Exponential Functions — AP® Precalculus Notes

In the last topic a logarithm answered the question "which exponent?". This page turns that into a statement about functions: an exponential function and the logarithm with the same base are inverse functions. That one fact explains their graphs, their domains and asymptotes, and the way each one changes.

Two functions that undo each other

Take $f(x) = b^x$ and $g(x) = \log_b x$ , with the same base $b > 0$ , $b \ne 1$ . If $f(a) = c$ , then $b^a = c$ , which means $\log_b c = a$ , so $g(c) = a$ . Each function sends the other's outputs back to its inputs. They are inverses: $g = f^{-1}$ .

Composing them gives the identity function:

$\log_b(b^x) = x$ for every real number $x$ ;
$b^{\log_b x} = x$ for every $x > 0$ (the logarithm needs a positive input).

The pair used most is $e^x$ and $\ln x$ : $\ln(e^x) = x$ and $e^{\ln x} = x$ .

Worked example 1. Evaluate $2^{\log_2 9}$ and $\ln e^{-4}$ .

$2^x$ undoes $\log_2 x$ , so $2^{\log_2 9} = 9$ .

$\ln$ undoes $e^x$ , so $\ln e^{-4} = -4$ .

The graphs

Inverse functions have graphs that are reflections across the line $y = x$ , and every feature trades places.

Feature	$y = b^x$ ( $b > 1$ )	$y = \log_b x$ ( $b > 1$ )
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Asymptote	horizontal, $y = 0$	vertical, $x = 0$
Key point	$(0, 1)$	$(1, 0)$
Another point	$(1, b)$	$(b, 1)$

The asymptote of the exponential, $\lim_{x \to -\infty} b^x = 0$ , becomes the behavior of the logarithm near $0$ : $\lim_{x \to 0^+} \log_b x$ $= -\infty$ .

How each one changes

An exponential function turns adding to the input into multiplying the output: $b^{x + k} = b^k \cdot b^x$ . For $f(x) = 2^x$ , each increase of $3$ in $x$ multiplies $f(x)$ by $2^3 = 8$ .

Its inverse does the reverse: multiplying the input of a logarithm adds to the output. Multiplying $x$ by $1{,}000 = 10^3$ adds $3$ to $\log x$ .

Worked example 2. A table gives $h(x) = \log_4 x$ at $x$ $= \frac{1}{16}, \frac{1}{4}, 1, 4, 16, 64$ . Describe the pattern.

Each input is $4$ times the one before, and the outputs are $-2, -1, 0, 1, 2, 3$ .

So equal factors of $4$ in the input give equal steps of $1$ in the output. Swap the rows and you get $4^x$ : equal steps of $1$ in the input give equal factors of $4$ in the output.

Inverting a stretched exponential

When an exponential has other steps around it, undo those steps first, in reverse order, and use the logarithm for the exponential step.

Worked example 3. Find the inverse of $f(x) = 2e^x$ .

Swap: $x = 2e^y$ .

Undo the factor $2$ : $e^y = \frac{x}{2}$ .

Undo $e$ : $y = \ln \frac{x}{2}$ . So $f^{-1}(x) = \ln \frac{x}{2}$ , defined for $x > 0$ , the range of $f$ .

Reading the inverse in context

If $N(t) = 500 \cdot 2^t$ counts bacteria after $t$ hours, the inverse answers "when?": $t = \log_2 \frac{N}{500}$ . The count $4{,}000$ is reached at $t = \log_2 8 = 3$ hours. The exponential takes a time and gives a count; its inverse takes a count and gives a time.