A2.2 Relate exponential and logarithmic functions as inverses

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Understanding how powers and logarithms undo each other

An exponential function answers a question about powers. A logarithmic function answers the matching question in reverse. For example, if a power of

3

gives

81

, a logarithm tells us which exponent was used. This lesson connects these functions as inverses. It uses tables, symbols, and one guided example to show how their inputs and outputs switch places.

What you will learn

Explain what it means for two functions to be inverses.
Relate an exponential function to its logarithmic inverse.
Use input-output pairs and a graph description to recognize inverse functions.
Check an inverse relationship by composing the functions in either order.

1. Prerequisite bridge: inputs, outputs, and powers

A function is a rule that assigns one output to each allowed input. You can think of it as a machine: you choose an input, apply the rule, and get an output. For instance, the rule

f(x)=3^x

gives the output

9

when the input is

2

, because

3^2=9

.

An exponent tells how many times the base is used as a factor. In

3^2

, the base is

3

and the exponent is

2

. An exponential function has its variable in the exponent, as in

3^x

. A logarithm names the exponent needed to produce a given positive value.

3^2=9

In $f(2)=9$ , the input is $2$ and the output is $9$ .
For a positive base other than $1$ , an exponential function can be paired with a logarithmic function.

2. Plain-language relationship: each function undoes the other

Two functions are inverses when each one undoes the action of the other. The exponential function

f(x)=3^x

asks, “What power of

3

is this input?” The matching logarithmic function asks, “What exponent on

3

gives this input?”

The notation is not needed to understand the relationship: if

3^2=9

, then would be the exponent that makes

3

become

9

. Written correctly, that exponent is . These paired statements show the same fact in two forms: a power statement and a logarithm statement.

More generally, for a base

a

with

a>0

and

a\ne1

, the exponential function

a^x

has inverse function

\log_a x

. The symbol

\log_a x

means the exponent that

a

must have to produce

x

. The input of a logarithm must be positive, because a positive base raised to a real exponent does not produce zero or a negative number.

An inverse reverses input and output. If the exponential function takes

2

to

9

, its inverse takes

9

back to

2

. This reversal is the key idea behind both the table and the graph.

a^x=y\iff\log_a y=x

The logarithm gives the exponent in an equivalent exponential statement.
The exponential function and its logarithmic inverse exchange inputs and outputs.
For base $a$ , use $a>0$ and $a\ne1$ .

3. Multiple representations: pairs, symbols, and graphs

A table makes the reversal visible. For

f(x)=3^x

, an input of

2

gives an output of

9

. The inverse relation therefore has input

9

and output

2

. Every ordered pair

(x,y)

for the exponential function becomes

(y,x)

for its inverse.

This coordinate switch also describes the graph. The graphs of inverse functions are mirror images across the line

y=x

. Reflecting a point

(x,y)

across that line gives the point

(y,x)

. The exponential graph passes through

(0,1)

because

a^0=1

. Its logarithmic inverse passes through

(1,0)

because

\log_a 1=0

.

The same relationship can be checked using composition. Composition means applying one function and then applying another to its output. If

f(x)=a^x

and

g(x)=\log_a x

, applying

g

after

f

returns the starting input. Applying

f

after

g

also returns the starting input, as long as the input to the logarithm is positive.

f(g(x))=x,\quad g(f(x))=x

Inverse pairs reverse the order of the coordinates.
Inverse graphs reflect across $y=x$ .
The exponential function has domain all real numbers and positive outputs; its logarithmic inverse has positive inputs and can produce any real output.

4. Applying the relationship

To move between exponential and logarithmic forms, identify the base, the output, and the exponent. In the statement

a^x=y

, the logarithmic form is

\log_a y=x

. The base stays the same. The power's result becomes the logarithm's input, and the exponent becomes the logarithm's value.

To identify an inverse function, reverse the input-output roles. Then check that the resulting rule undoes the original. For exponential and logarithmic functions with the same valid base, this gives

\log_a(a^x)=x

and

a^{\log_a x}=x

for positive

x

.

This relationship is useful whenever an unknown appears as an exponent. A logarithm expresses that unknown as a value. The important point here is not a new calculation method; it is recognizing that the logarithmic function reverses the exponential function.

\log_a(a^x)=x,\quad a^{\log_a x}=x

Keep the same base when rewriting between exponential and logarithmic forms.
Check inverse functions by confirming that each order of application returns the starting input.

Input-output pairs for a function and its inverse

Exponential input	Exponential output	Inverse input	Inverse output
0	1	1	0
1	3	3	1
2	9	9	2

Worked example

Finding and checking the inverse of an exponential function

Let

f(x)=3^x

. State its inverse, use input-output pairs to show the reversal, and check the relationship in both orders.

Identify the base and output
The function has base $3$ . A useful pair comes from choosing $x=2$ : the exponential rule gives an output of $9$ .
$f(2)=3^2=9$
Reverse the pair
An inverse switches the input and output. Since the original function takes $2$ to $9$ , its inverse must take $9$ to $2$ . The logarithm names the exponent needed to produce $9$ from base $3$ .
$(2,9)\longleftrightarrow(9,2),\quad\log_3 9=2$
Write the inverse rule
Replace the exponential relationship with its logarithmic form. The base remains $3$ , and the input and exponent roles are exchanged. Thus the inverse function is $f^{-1}(x)=\log_3 x$ , defined for positive $x$ .
$f^{-1}(x)=\log_3 x$
Check both orders
Applying the logarithm to an exponential output recovers the original exponent. Applying the exponential to a logarithm input recovers that positive input. These checks confirm that the rules undo each other.
$\log_3(3^x)=x,\quad 3^{\log_3 x}=x\ (x>0)$

Answer: The inverse of

f(x)=3^x

is

f^{-1}(x)=\log_3 x

. The pair

(2,9)

becomes

(9,2)

, and the two composition checks return the starting input.

Check: The exponential function accepts any real input and produces a positive output. The inverse therefore accepts positive inputs and produces real outputs, matching the logarithm's required input.

Common mistakes and how to avoid them

Treating

\log_a y

as multiplication by

a

.

Correction: A logarithm is an exponent. The expression

\log_a y

asks which exponent on

a

produces

y

.

Changing the base when switching between exponential and logarithmic forms.

Correction: Keep the same base. In

a^x=y

, the equivalent logarithmic form is

\log_a y=x

.

Reversing the base and exponent instead of reversing the input and output.

Correction: Keep the base fixed and exchange the exponent with the resulting value. For example,

3^2=9

corresponds to

\log_3 9=2

.

Allowing zero or a negative number as the input to a real logarithm.

Correction: The input of

\log_a x

must be positive. The matching exponential function produces positive outputs only.

Lesson summary

An exponential function and its logarithmic function with the same valid base are inverses.
The statement $a^x=y$ is equivalent to $\log_a y=x$ .
Inverse functions reverse input-output pairs, and their graphs reflect across $y=x$ .
Applying either function and then its inverse returns the starting input, within the functions' domains.

Check your understanding

Question 1

Which logarithmic statement is equivalent to

5^3=125

?

$\log_5 125=3$
$\log_{125} 5=3$
$\log_3 125=5$
correctIndex': 0,

Show answer and explanation

\log_5 125=3

The base remains

5

, the result becomes the logarithm's input, and the exponent becomes its value.

Question 2

If an exponential function has the point

(1,4)

, which point belongs to its inverse?

$(1,4)$
$(4,1)$
$(4,4)$
correctIndex': 1,

Show answer and explanation

(4,1)

An inverse switches the coordinates, so the point

(1,4)

becomes

(4,1)

.

Question 3

What is the domain of the inverse of

f(x)=2^x

?

All real numbers
Positive real numbers
Non-negative real numbers
correctIndex': 1,

Show answer and explanation

Positive real numbers

The range of

2^x

is the positive real numbers, and the inverse uses that range as its domain.

Key terms

Function: A rule that assigns one output to each allowed input.
Exponential function: A function in which the variable appears as an exponent, such as $a^x$ .
Logarithm: The exponent that a specified base must have to produce a given positive number.
Inverse functions: Two functions that undo each other by reversing input and output.
Composition: Applying one function and then applying another function to the result.
Domain: The set of allowed input values for a function.
Range: The set of output values a function can produce.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A2.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A2.2 · Relate exponential and logarithmic functions as inverses

Ontario Grade 12 Mathematics

Understanding how powers and logarithms undo each other

What you will learn

1. Prerequisite bridge: inputs, outputs, and powers

2. Plain-language relationship: each function undoes the other

3. Multiple representations: pairs, symbols, and graphs

4. Applying the relationship

Input-output pairs for a function and its inverse

Finding and checking the inverse of an exponential function

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson