Recognize equivalent exponential and logarithmic expressions

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

MHF4U study topic A3.1

An exponential expression shows a base raised to an exponent. A logarithmic expression asks what exponent on a base produces a given result. These are two ways to describe the same relationship. For example,

2^3=8

says that

2

raised to the exponent

3

equals

8

. The matching logarithmic statement is

\\log_2 8=3

: the exponent needed on

2

to produce

8

is

3

. In this lesson, you will learn to recognize and rewrite these equivalent expressions. You will use positive bases other than

1

, and positive results inside logarithms.

What you will learn

Explain how an exponential statement and a logarithmic statement can express the same relationship.
Translate between exponential and logarithmic forms while keeping the base, exponent, and result in the correct positions.
Check whether a proposed logarithmic expression is defined and whether it matches an exponential statement.

1. Prerequisite bridge: bases, powers, and exponents

A power has a base and an exponent. In

5^2

, the base is

5

and the exponent is

2

. The expression means

5\\times5

, so its value is

25

.

An exponential equation can state the value of a power, as in

5^2=25

. The base and the result are both visible, and the exponent tells how many times the base is used as a factor. A logarithm is a way to name that exponent.

Before writing a logarithm, check the base and result. In the relationship used here, the base must be positive and cannot equal

1

. The result must be positive. These conditions make the logarithmic expression meaningful in this lesson.

a^b=c

In $a^b=c$ , $a$ is the base, $b$ is the exponent, and $c$ is the result.
For the forms in this lesson, $a>0$ , $a\\ne1$ , and $c>0$ .

2. The same relationship in two forms

The logarithmic statement

\\log_a c=b

means: “The exponent on

a

that gives

c

is

b

.” The small subscript

a

is the base of the logarithm. The number

c

is called the argument; it is the value inside the logarithm. The answer,

b

, is the exponent being described.

To translate an exponential statement into logarithmic form, keep the base, result, and exponent in their roles. The base becomes the logarithm’s subscript, the result becomes its argument, and the exponent becomes the value on the other side of the equals sign. To translate back, the exponent in the logarithmic statement becomes the exponent in the power.

For instance,

3^4=81

and

\\log_3 81=4

express the same fact. Neither form changes the numbers or the relationship; only the way the relationship is written changes.

a^b=c\ \\Longleftrightarrow\ \\log_a c=b

Exponential form: base raised to exponent equals result.
Logarithmic form: logarithm with that base and result as argument equals the exponent.
Read a logarithm as a question about an exponent.

3. Use a table to keep the parts aligned

When expressions look unfamiliar, label the parts before rewriting them. The table below uses the same relationship in each row. The labels help prevent swapping the exponent and the result.

A logarithm is not defined in these forms when its argument is zero or negative. Also, a base of

1

cannot produce different positive results by changing its exponent, so it is excluded. These restrictions are useful checks when deciding whether two forms can match.

Match the base with the logarithm’s subscript.
Match the exponential result with the logarithm’s argument.
Match the exponential exponent with the value of the logarithm.

Matching the parts of equivalent expressions

Exponential form	Role	Logarithmic form
$a$	Base	Subscript $a$
$b$	Exponent	Value $b$
$c$	Result	Argument $c$
$a^b=c$	Full relationship	$\\log_a c=b$

Worked example

Translate and verify a relationship

Rewrite

4^3=64

in logarithmic form. Then rewrite

\\log_5 125=3

in exponential form. Explain how the positions of the numbers determine each rewrite.

Identify the roles
In $4^3=64$ , the base is $4$ , the exponent is $3$ , and the result is $64$ . In a logarithmic form, the base becomes the subscript, the result becomes the argument, and the exponent is the value of the logarithm.
$4^3=64$
Write the matching logarithm
Place the base $4$ as the subscript and the result $64$ inside the logarithm. The exponent $3$ goes on the other side of the equals sign. The statement says that $4$ to the third power gives $64$ .
$\\log_4 64=3$
Identify the roles in the logarithm
In $\\log_5 125=3$ , the base is $5$ , the argument is $125$ , and the logarithm’s value is $3$ . To return to exponential form, use the base raised to that value, with the argument as the result.
$\\log_5 125=3$
Write the matching power
The base $5$ raised to the exponent $3$ equals $125$ . This matches the meaning of the given logarithm because $5^3$ is $125$ .
$5^3=125$

Answer: The equivalent forms are

4^3=64

and

\\log_4 64=3

, and

\\log_5 125=3

and

5^3=125

.

Check: Evaluate each power:

4^3=4\\times4\\times4=64

and

5^3=5\\times5\\times5=125

. Both translations preserve the same base, exponent, and result.

Common mistakes and how to avoid them

Putting the exponent inside the logarithm and the result outside, such as rewriting

2^5=32

as

\\log_2 5=32

.

Correction: The result becomes the argument, and the exponent becomes the logarithm’s value. The correct form is

\\log_2 32=5

.

Changing the base during a rewrite, such as claiming

3^4=81

is equivalent to

\\log_4 81=3

.

Correction: Keep the original base as the logarithm’s subscript. The matching form is

\\log_3 81=4

.

Writing a logarithm with a zero or negative argument in this relationship.

Correction: The argument must be positive. For example,

\\log_2 0

and

\\log_2(-8)

are not defined as real logarithms.

Assuming every positive number can be a base, including

1

.

Correction: The base must not equal

1

. A power with base

1

always has value

1

, so it cannot represent the general relationship described here.

Lesson summary

An exponential equation and a logarithmic equation can describe the same base, exponent, and result.
In $a^b=c$ , the equivalent logarithmic form is $\\log_a c=b$ .
To translate, preserve the roles: base to subscript, result to argument, and exponent to logarithm value.
For these logarithmic forms, the base is positive and not $1$ , and the argument is positive.

Check your understanding

Question 1

Which logarithmic expression is equivalent to

7^2=49

?

$\\log_7 49=2$
$\\log_2 49=7$
$\\log_7 2=49$
$\\log_{49} 7=2$

Show answer and explanation

\\log_7 49=2

The base

7

becomes the subscript, the result

49

becomes the argument, and the exponent

2

becomes the value.

Question 2

Which exponential equation is equivalent to

\\log_3 81=4

?

$3^4=81$
$4^3=81$
$3^{81}=4$
$81^3=4$

Show answer and explanation

3^4=81

The logarithm says that the exponent on base

3

needed to produce

81

is

4

, so the equivalent power is

3^4=81

.

Question 3

Which expression is not a valid real logarithm in the form used in this lesson?

$\\log_2 16$
$\\log_5 1$
$\\log_1 8$
$\\log_4 9$

Show answer and explanation

\\log_1 8

A logarithm’s base must be positive and cannot equal

1

. The other expressions have positive arguments and valid bases.

Key terms

Base: The number that is raised to an exponent in a power. In $a^b$ , the base is $a$ .
Exponent: The number that tells the power of the base. In $a^b$ , the exponent is $b$ .
Logarithm: A way to express the exponent needed on a base to produce a given positive value.
Argument: The value inside a logarithm. In $\\log_a c$ , the argument is $c$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A3.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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