A1.3 Connect logarithmic and exponential equations

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Reading each form as a statement about the same exponent

A logarithm answers a question about an exponent. For example, asking “What exponent on 2 gives 8?” can be written as either

2^3=8

or

\log_2 8=3

. These are not two different calculations. They are two ways to record the same relationship. In this lesson, you will connect the forms, use the connection to solve an equation, and check that the result makes sense.

What you will learn

Explain how a logarithmic equation and an exponential equation can express the same relationship.
Rewrite an equation from logarithmic form to exponential form, and from exponential form to logarithmic form.
Use the connection to solve an equation and check that the answer is allowed.

1. Prerequisite bridge: powers and bases

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

3^2=9

, the base is

3

and the exponent is

2

. The equation says that raising

3

to the power

2

gives

9

.

A logarithm reverses this question. The expression

\log_3 9

asks: “What exponent on

3

gives

9

?” Since

3^2=9

, the answer is

2

. Thus,

\log_3 9=2

.

In

\log_b a=c

, the base is

b

, the argument is

a

, and the logarithm's value is

c

. The argument is the number inside the logarithm. For real logarithms, the base must be positive and not equal to

1

, and the argument must be positive. These conditions matter when solving equations.

\log_b a=c\iff b^c=a

An exponent tells the result of raising a base to a power.
A logarithm asks which exponent produces a given result.
For $\log_b a$ , require $b>0$ , $b\ne 1$ , and $a>0$ .

2. The connection in words, numbers, and symbols

The symbol

\iff

means “if and only if.” Here it marks two statements that mean exactly the same thing, as long as the base and argument meet the required conditions. To change from logarithmic form to exponential form, the base stays the base, the logarithm's value becomes the exponent, and the argument becomes the result.

Consider

\log_5 125=3

. The base is

5

, the value is

3

, and the argument is

125

. The matching power statement is

5^3=125

. In the other direction,

4^3=64

can be written as

\log_4 64=3

.

This connection applies when the unknown is in different places. In

\log_2 x=5

, the logarithm asks which power of

2

equals

x

. In

3^x=20

, the unknown is the exponent, so the logarithmic form is

x=\log_3 20

. The second equation does not claim that

x

is a whole number; the logarithm names the exact exponent.

\log_b a=c\iff b^c=a

In the conversion, the base remains the base.
The logarithm's value becomes the exponent.
The argument becomes the result of the power.

3. Solving equations and checking restrictions

When a logarithmic equation has one logarithm equal to a number, rewrite it as a power equation. Then solve the resulting equation using familiar algebra. When an exponential equation has a variable exponent, rewrite it as a logarithmic equation if that gives a direct expression for the unknown.

For example, changing

\log_2(x+3)=4

to exponential form gives

2^4=x+3

. This turns the logarithm into an ordinary equation. Solve for

x

, then check the original logarithm's argument. Since

x+3

must be positive, a proposed solution that makes

x+3

zero or negative is not allowed.

A check has two parts. First, substitute the candidate into the original equation and confirm the equality. Second, confirm that each logarithm has a positive argument and a valid base. A value that solves a rearranged equation is not a solution if it makes the original logarithm undefined.

Keep the order of the parts clear. In

\log_b a=c

, the base

b

is the base of the power, while

a

is the result. Reversing these roles changes the equation. For instance,

\log_2 8=3

corresponds to

2^3=8

, not

8^2=3

.

\log_b(a)=c\iff b^c=a

Convert the equation before solving when that makes the unknown easier to isolate.
Check the result in the original equation.
Every logarithm's argument must be positive.

4. Applying the connection

The two forms are useful for different questions. Exponential form makes it easy to calculate a result when the exponent is known. Logarithmic form names the exponent when the result is known. For example,

2^5=32

quickly gives the value of a power, while

\log_2 32=5

directly states the exponent.

When solving, choose the form that exposes the unknown. If the unknown is inside the logarithm as part of its argument, converting to exponential form often produces an equation that can be solved with algebra. If the unknown is the exponent, logarithmic form can state its exact value.

The connection is also a way to check whether a proposed rewrite is correct. Read the logarithmic equation as a question: “The base raised to what exponent equals the argument?” If the power statement answers that question, the conversion is consistent.

b^c=a\iff \log_b a=c

Use powers to evaluate a known exponent.
Use logarithms to name an unknown exponent.
Read the base, exponent, and result carefully when checking a conversion.

Two forms, one relationship

Logarithmic form	Question it answers	Matching exponential form
$\log_5 125=3$	What power of $5$ gives $125$ ?	$5^3=125$
$\log_4 64=3$	What power of $4$ gives $64$ ?	$4^3=64$
$\log_3 20=x$	What power of $3$ gives $20$ ?	$3^x=20$

Worked example

Solve a logarithmic equation by converting forms

Solve

\log_2(x+3)=4

and check the result.

Identify the parts
The base is $2$ , the logarithm's value is $4$ , and its argument is $x+3$ . Rewrite the statement so that raising $2$ to the power $4$ gives the argument.
$2^4=x+3$
Evaluate the power
Since $2^4=16$ , the equation states that $x+3$ equals $16$ . This is now a linear equation.
$16=x+3$
Isolate the variable
Subtract $3$ from both sides to leave $x$ by itself.
$x=13$
Check the original equation
The argument is $13+3=16$ , which is positive. Also, $2^4=16$ , so the logarithm of $16$ with base $2$ is $4$ . The candidate satisfies the original equation and its domain restriction.
$\log_2(13+3)=\log_2 16=4$

Answer:

x=13

Check: The original logarithm is defined because its argument is

16>0

, and its value is

4

.

Common mistakes and how to avoid them

Writing the argument as the base when changing forms.

Correction: Keep the base unchanged. In

\log_b a=c

, the matching power is

b^c=a

.

Treating the logarithm's value as the result of the power.

Correction: The logarithm's value becomes the exponent. The argument becomes the result.

Keeping a candidate even though it makes a logarithm's argument zero or negative.

Correction: Check the original equation. Every logarithm's argument must be positive.

Assuming the unknown exponent must be a whole number.

Correction: An exponent can be a non-integer. A logarithm can represent the exact value of that exponent.

Lesson summary

A logarithmic equation and its matching exponential equation express the same relationship.
In $\log_b a=c$ , the equivalent power statement is $b^c=a$ .
Use the form that makes the unknown easier to identify or solve.
Check solutions in the original equation and make sure every logarithm's argument is positive.

Check your understanding

Question 1

Which exponential equation matches

\log_3 81=4

?

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

Show answer and explanation

3^4=81

The base stays

3

, the logarithm's value becomes the exponent

4

, and the argument

81

is the result.

Question 2

Which logarithmic equation matches

5^x=25

?

$\log_5 25=x$
$\log_{25} 5=x$
$\log_5 x=25$
$\log_x 5=25$

Show answer and explanation

\log_5 25=x

The base is

5

, the result is

25

, and the unknown exponent is the logarithm's value.

Question 3

Solve

\log_4(x-1)=2

. Which value is valid?

$x=17$
$x=5$
$x=0$
$x=15$

Show answer and explanation

x=17

Convert to

4^2=x-1

, so

16=x-1

and

x=17

. Its argument is

17-1=16>0

, and

\log_4 16=2

.

Key terms

Base: The number being raised to a power; in $\log_b a$ , it is $b$ .
Exponent: The number that tells which power of the base is used.
Argument: The number or expression inside a logarithm; in $\log_b a$ , it is $a$ .
Logarithm: The exponent that makes a given base equal a specified result.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A1.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A1.3 · Connect logarithmic and exponential equations

Ontario Grade 12 Mathematics

Reading each form as a statement about the same exponent

What you will learn

1. Prerequisite bridge: powers and bases

2. The connection in words, numbers, and symbols

3. Solving equations and checking restrictions

4. Applying the connection

Two forms, one relationship

Solve a logarithmic equation by converting forms

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