AP®MathsCollege BoardPrecalculusStudy GuidesExponential & Logarithmic FunctionsLogarithmic FunctionsLogarithmic Functions as Inverses of Exponential Functions

Logarithmic Functions as Inverses of Exponential Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Logarithmic functions as inverses of exponential functions

The logarithmic function $f (x) = \log_{b} x$
- and the exponential function $g (x) = b^{x}$
  - where $b > 0$ and $b \neq 1$
- are inverse functions of each other
This means they "undo" each other:
- $\log_{b} (b^{x}) = x$
  - taking the log of an exponential with the same base returns the exponent
- $b^{l o g_{b} x} = x$
  - raising a base to a logarithm with the same base returns the original value inside the logarithm
More formally, if $f (x) = \log_{b} x$ and $g (x) = b^{x}$
- then $f (g (x)) = g (f (x)) = x$

How do the input-output behaviors of logarithmic and exponential functions compare?

Exponential and logarithmic functions handle change in opposite ways
In an exponential function
- when input values increase by equal amounts (additively)
- the output values change by equal ratios (multiplicatively)
In a logarithmic function, the relationship is reversed
- when input values change by equal ratios (multiplicatively)
- the output values increase by equal amounts (additively)
E.g. for $g (x) = 2^{x}$
- increasing $x$ by 1 (additive change)
- always doubles the output (multiplicative change)
For $f (x) = \log_{2} x$
- doubling $x$ (multiplicative change)
- always increases the output by 1 (additive change)

$x$	$g (x) = 2^{x}$	$x$	$f (x) = \log_{2} x$
0	1	1	0
1	2	2	1
2	4	4	2
3	8	8	3
4	16	16	4

Notice that the tables are mirrors of each other
- The inputs and outputs are swapped
- This is exactly the inverse relationship at work

How do ordered pairs relate between the two functions?

If $(s, t)$ is a point on the graph of the exponential function $g (x) = b^{x}$
- then $(t, s)$ is a point on the graph of the logarithmic function $f (x) = \log_{b} x$
This follows directly from the inverse relationship
- if $b^{s} = t$ , then $\log_{b} t = s$
E.g. since $2^{3} = 8$
- the point $(3, 8)$ is on the graph of $g (x) = 2^{x}$
- and the point $(8, 3)$ is on the graph of $f (x) = \log_{2} x$

Graphs of logarithmic functions

The graph of $f (x) = \log_{b} x$ is the reflection of the graph of $g (x) = b^{x}$ over the line $y = x$
- This is true for any base $b$ (with $b > 0$ and $b \neq 1$ )
- It follows from the general principle that the graph of any function and its inverse are reflections of each other over $y = x$
Note how features of the graphs swap under reflection:
- The exponential function has a horizontal asymptote ( $y = 0$ )
  - the logarithmic function has a vertical asymptote ( $x = 0$ )
- The exponential function passes through $(0, 1)$
  - the logarithmic function passes through $(1, 0)$
- The exponential function has domain all real numbers and range $y > 0$
  - the logarithmic function has domain $x > 0$ and range all real numbers
- These graph features are true for any base $b$ (with $b > 0$ and $b \neq 1$ )

Graph showing lines y=x, y=2^x, and y=log_2(x). Asymptotes at y=0 and x=0, points at (0,1) and (1,0). — **Graphs of exponential and logarithmic functions**

Worked Example

The exponential function $g (x) = 3^{x}$ and the logarithmic function $f (x) = \log_{3} x$ are inverse functions.

(a) Without using a calculator, complete the tables of values for $g (x) = 3^{x}$ and $f (x) = \log_{3} x$ .

$x$	$- 2$	$- 1$	0	1	2	3
$g (x) = 3^{x}$

$x$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9	27
$f (x) = \log_{3} x$

Answer:

You can use rules of exponents to complete the table for $g$

$3^{- 2} = \frac{1}{3^{2}} = \frac{1}{9} 3^{- 1} = \frac{1}{3^{1}} = \frac{1}{3} 3^{0} = 1$

$3^{1} = 3 3^{2} = 9 3^{3} = 27$

$x$	$- 2$	$- 1$	0	1	2	3
$g (x) = 3^{x}$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9	27

To find the table for $f (x) = \log_{3} x$ , swap the inputs and outputs in the table for $g$

This works because the functions are inverses

$x$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9	27
$f (x) = \log_{3} x$	$- 2$	$- 1$	0	1	2	3

(b) Show that $f (g (x)) = x$ by evaluating $f (g (2))$ .

Answer:

Start with the value of $g (2)$

$g (2) = 3^{2} = 9$

Substitute that into $f (x)$ , and find the value

$f (g (2)) = f (9) = \log_{3} 9 = 2$

So the output is equal to the input

The value of $f (g (2))$ equals the original input (2), illustrating the fact that $f (g (x)) = x$

Rewriting exponential functions using logarithms

How can logarithms be used to rewrite exponential expressions?

Logarithms provide a way to change the base of an exponential expression
- This can be useful for comparing or simplifying expressions
The key identity is $b^{x} = c^{(\log_{c} b) (x)}$
- This works because $b = c^{\log_{c} b}$
  - I.e., by the inverse properties of logarithms and exponentials
  - $c^{□}$ 'cancels' $\log_{c}$
- Therefore $b^{x} = {(c^{\log_{c} b})}^{x} = c^{(\log_{c} b) (x)}$

Examiner Tips and Tricks

The identity $b^{x} = c^{(\log_{c} b) (x)}$ may look complicated

But it follows directly from the fact that $b$ can be written as $c^{\log_{c} b}$
If you can remember that one fact, you can derive the rest

Also be careful with the notation here!

$(\log_{c} b) (x)$ means
- the number $\log_{c} b$
- times the input variable $x$
It does not mean $\log_{c} (b x)$
- which is the logarithm to base $c$ of $b x$

This means any exponential function can be rewritten using any base you choose
- With the notation used above, you can use the identity to rewrite an exponential with base $b$ as an exponential with base $c$
E.g. rewrite $3^{x}$ using the base $e$
- Use the identity
  - $3^{x} = e^{(\ln 3) (x)}$
    - since $\ln 3 = \log_{e} 3$
- Or with the brackets expanded
  - $3^{x} = e^{x \ln 3}$
E.g. rewrite $5^{x}$ using base 10
- Using the identity
  - $5^{x} = 10^{(\log_{10} 5) (x)}$
- Or with the brackets expanded
  - $5^{x} = 10^{x \log 5}$

Examiner Tips and Tricks

A very common application is converting any exponential to base $e$ .

For this the formula can be written more simply as $b^{x} = e^{x \ln b}$

When rewriting exponential expressions in a free response question on the exam, always show the intermediate steps clearly.

Worked Example

Rewrite each of the following exponential expressions in the specified form.

(a) Rewrite $7^{x}$ in the form $e^{k x}$ , where $k$ is a constant. Express $k$ as a decimal approximation to three decimal places.

Answer:

Use the identity $b^{x} = e^{(\ln b) (x)}$ with $b = 7$

$7^{x} = e^{(\ln 7) (x)}$

Use a calculator to find the value of $\ln 7$

$k = \ln 7 = 1.945910 . . .$

Round $k$ to 3 decimal places

$7^{x} = e^{1.946 x}$

(b) Rewrite $4^{x}$ in the form $2^{m x}$ , where $m$ is a constant.

Answer:

The easiest way to do this is to substitute $2^{2}$ in place of 4

and then use rules of exponents

$4^{x} = (2^{2})^{x} = 2^{2 x}$

That is in the form you are looking for, with $m = 2$

You could also work this out using the identity $b^{x} = c^{(\log_{c} b) (x)}$

with $b = 4$ and $c = 2$

$4^{x} = 2^{(\log_{2} 4) (x)} = 2^{(2) (x)} = 2^{2 x}$

That follows since $\log_{2} 4 = 2$

$4^{x} = 2^{2 x}$

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Logarithmic Functions as Inverses of Exponential Functions (College Board AP® Precalculus): Study Guide

Logarithmic functions as inverses of exponential functions

How do the input-output behaviors of logarithmic and exponential functions compare?

How do ordered pairs relate between the two functions?

Graphs of logarithmic functions

Worked Example

Rewriting exponential functions using logarithms

How can logarithms be used to rewrite exponential expressions?

Examiner Tips and Tricks

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis

Logarithmic Functions as Inverses of Exponential Functions (College Board AP® Precalculus): Study Guide

Logarithmic functions as inverses of exponential functions

How are exponential and logarithmic functions related?

How do the input-output behaviors of logarithmic and exponential functions compare?

How do ordered pairs relate between the two functions?

Graphs of logarithmic functions

How is the graph of a logarithmic function related to the graph of an exponential function?

Rewriting exponential functions using logarithms

How can logarithms be used to rewrite exponential expressions?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis