Connect equivalent exponential equations written with different bases

Ontario Grade 11 Mathematics

Exponential Functions

MCR3U — B2.4: Writing the Same Exponential Relationship Using a Different Base

You already know how to evaluate powers like

2^3 = 8

and

4^2 = 16

. But what happens when two expressions look completely different yet produce the same value? For example,

2^6

and

4^3

both equal

64

, even though one uses base

2

and the other uses base

4

. This lesson is all about that connection. You will learn how to deliberately rewrite an exponential expression using a different — but equivalent — base, and then use that skill to connect or solve exponential equations. Everything here works by recognising how bases relate to each other as powers, with no tools beyond what you have already seen in this course.

What you will learn

Recognize when two exponential expressions with different bases represent the same value.
Rewrite a power using a new base by identifying the correct exponent.
Connect equivalent exponential equations by expressing both sides with a common base.
Solve for an unknown exponent by matching bases on both sides of an equation.

Prerequisite Bridge: Powers and Exponent Laws

Before connecting different bases, make sure the following ideas feel comfortable. A power is written as

b^n

, where

b

is the base and

n

is the exponent. The value

b^n

means

b

multiplied by itself

n

times. For example,

3^4 = 3 \times 3 \times 3 \times 3 = 81

.

The power-of-a-power law states

(b^m)^n = b^{m \times n}

. This law is the engine of today's lesson. For example,

(2^3)^2 = 2^{3 \times 2} = 2^6 = 64

. Notice that

8^2 = 64

as well, so

2^6

and

8^2

are simply two names for the same number.

One more key idea: any base can itself be written as a power of a smaller base, as long as you can find the right relationship. For instance,

9 = 3^2

, so any power of

9

can be rewritten as a power of

3

. Spotting these relationships is the main skill you will practise today.

(b^m)^n = b^{mn}

A power $b^n$ means repeated multiplication of the base $b$ .
Power-of-a-power law: $(b^m)^n = b^{mn}$ .
If $b = c^k$ , then $b^n = (c^k)^n = c^{kn}$ , which rewrites base $b$ as base $c$ .

Rewriting a Base as a Power of Another Base

The central idea of this lesson is: if you want to rewrite

b^n

using a new base

c

, first ask yourself, 'Is

b

a power of

c

?' If

b = c^k

for some rational number

k

, you can substitute and use the power-of-a-power law.

Consider the bases

8

and

2

. Because

8 = 2^3

, any power of

8

can be converted to a power of

2

:

8^n = (2^3)^n = 2^{3n}

. So

8^5 = 2^{15}

and

8^{-2} = 2^{-6}

. The value does not change — only the way it is written changes.

You can also go in the other direction. To write a power of

2

as a power of

8

, notice that

2 = 8^{1/3}

because the cube root of

8

is

2

. So

2^n = (8^{1/3})^n = 8^{n/3}

. For example,

2^{12} = 8^4

. Check:

8^4 = 4096

and

2^{12} = 4096

. ✓

This works for any pair of bases where one is a perfect power of the other. Common pairs to recognise are:

4

and

2

(since

4 = 2^2

),

8

and

2

(since

8 = 2^3

),

9

and

3

(since

9 = 3^2

),

27

and

3

(since

27 = 3^3

),

25

and

5

(since

25 = 5^2

), and

\frac{1}{4}

and

2

(since

\frac{1}{4} = 2^{-2}

).

b^n = (c^k)^n = c^{kn}

To rewrite $b^n$ in base $c$ , find $k$ such that $b = c^k$ , then $b^n = c^{kn}$ .
The value of the expression never changes — only its written form does.
Recognise common base pairs: $(4,2)$ , $(8,2)$ , $(9,3)$ , $(27,3)$ , $(25,5)$ .

Connecting Equivalent Exponential Equations

Two exponential equations are called equivalent if they describe exactly the same relationship, just written with different bases. For example, the equation

4^x = 64

and the equation

2^{2x} = 2^6

are equivalent — they share the same solution and the same underlying meaning, but the first uses base

4

and the second uses base

2

.

The method to connect them: rewrite every term in the equation so that all terms share one common base. Once both sides are written with the same base, you can compare exponents directly. This is valid because if

c^A = c^B

and

c > 0

with

c \neq 1

, then

A = B

. In plain language: equal powers with the same positive base (not equal to

1

) must have equal exponents.

Here is the general process. First, pick a common base — usually the smallest positive integer base that works for every term. Second, rewrite each term using that base and the power-of-a-power law. Third, set the exponents equal to each other. Fourth, solve the resulting equation, which will be linear or straightforward to handle using only the algebra skills from this course.

c^A = c^B \Rightarrow A = B

Two exponential equations are equivalent when they have the same solution and describe the same relationship.
Write both sides with a single common base, then compare exponents.
If $c^A = c^B$ with $c > 0$ and $c \neq 1$ , then $A = B$ .

Fractional and Negative Bases in Equivalent Forms

The same technique extends to cases where the base in the equation is a fraction or involves a negative exponent. Recall that

\frac{1}{b^n} = b^{-n}

, so

\left(\frac{1}{3}\right)^x = 3^{-x}

. This means an equation written with base

\frac{1}{3}

can always be rewritten with base

3

by negating the exponent.

For example, consider

\left(\frac{1}{9}\right)^x = 27

. Since

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

and

27 = 3^3

, the equation becomes

3^{-2x} = 3^3

. Setting exponents equal gives

-2x = 3

, so

x = -\frac{3}{2}

.

These forms look more intimidating, but the process is identical: find the common base, apply the power-of-a-power law, and equate exponents. The key is being systematic and taking one step at a time.

\left(\frac{1}{b}\right)^n = b^{-n}

$\frac{1}{b^n} = b^{-n}$ : a fraction in the base can always be written with a negative exponent.
Rewrite fractional bases using their equivalent integer base with a negative exponent, then apply the power-of-a-power law.
The solving step is always the same: equate the exponents once all bases match.

Checking Your Answer

Once you find a solution, always substitute it back into the original equation to verify. This habit catches sign errors and arithmetic slips before they cost you marks.

For the equation

\left(\frac{1}{9}\right)^x = 27

with solution

x = -\frac{3}{2}

: substitute to get

\left(\frac{1}{9}\right)^{-3/2}

. A negative exponent flips the fraction, giving

9^{3/2}

. The exponent

\frac{3}{2}

means take the square root first, then cube the result:

\sqrt{9} = 3

, then

3^3 = 27

. Since

27 = 27

, the answer is confirmed. ✓

Also keep the base condition in mind. The rule 'equal exponents imply equal bases' only works when the base is positive and not equal to

1

. If the base were

1

, then

1^A = 1^B = 1

for every choice of

A

and

B

, so you could not draw any conclusion about the exponents. This situation will not arise with the bases used in this course, but understanding why the condition exists helps you see why the method is trustworthy.

Always substitute your answer back into the original equation to verify.
The rule $c^A = c^B \Rightarrow A = B$ requires $c > 0$ and $c \neq 1$ .
Checking is not optional — it is part of a complete solution.

Common Base Relationships to Recognise

Expression	Common Base	Rewritten as a Power	Example Conversion
$4$	$2$	$2^2$	$4^3 = 2^6$
$8$	$2$	$2^3$	$8^4 = 2^{12}$
$9$	$3$	$3^2$	$9^x = 3^{2x}$
$27$	$3$	$3^3$	$27^2 = 3^6$
$25$	$5$	$5^2$	$25^x = 5^{2x}$
$\frac{1}{4}$	$2$	$2^{-2}$	$\left(\frac{1}{4}\right)^x = 2^{-2x}$
$\frac{1}{9}$	$3$	$3^{-2}$	$\left(\frac{1}{9}\right)^x = 3^{-2x}$

Worked example

Connecting Two Forms of the Same Equation

Show that the equation

4^x = 32

can be written equivalently as

2^{2x} = 2^5

, and then find the value of

x

.

Identify a common base
Both $4$ and $32$ can be written as powers of $2$ . Check: $4 = 2^2$ and $32 = 2^5$ . So $2$ is the common base to use throughout.
$4 = 2^2, \ 32 = 2^5$
Rewrite the left side using base 2
Replace $4$ with $2^2$ on the left side of $4^x = 32$ . Then apply the power-of-a-power law by multiplying the exponents: the outer exponent $x$ multiplies the inner exponent $2$ .
$(2^2)^x = 2^{2x}$
Write the full equation in base 2
Replacing both sides with their base- $2$ forms gives the equivalent equation. This confirms the two forms are connected — they are the same equation written differently.
$2^{2x} = 2^5$
Equate the exponents
Both sides now have the same base $2$ , and since $2 > 0$ with $2 \neq 1$ , the exponents must be equal to each other.
$2x = 5$
Solve for x
Divide both sides by $2$ to isolate $x$ .
$x = \frac{5}{2}$

Answer:

x = \frac{5}{2}

Check: Substitute

x = \frac{5}{2}

into the original equation:

4^{5/2} = (\sqrt{4})^5 = 2^5 = 32

. ✓

Worked example

Solving an Equation with a Fractional Base

Solve

\left(\frac{1}{27}\right)^{x} = 9

.

Find a common base for both sides
Notice that $27 = 3^3$ and $9 = 3^2$ . Both sides of the equation can be expressed using base $3$ .
$27 = 3^3, \ 9 = 3^2$
Rewrite the left-side base as a power of 3
Since $27 = 3^3$ , the fraction $\frac{1}{27}$ equals $3^{-3}$ , because a reciprocal produces a negative exponent.
$\frac{1}{27} = 3^{-3}$
Apply the power-of-a-power law to the left side
Replace $\frac{1}{27}$ with $3^{-3}$ inside the original expression, then multiply the exponents: $-3$ times $x$ gives $-3x$ .
$(3^{-3})^x = 3^{-3x}$
Write the full equation in base 3
Replace both sides with their base- $3$ forms. The equation is now written entirely in base $3$ .
$3^{-3x} = 3^{2}$
Equate the exponents
Since both sides share base $3$ and $3 > 0$ with $3 \neq 1$ , the exponents must be equal.
$-3x = 2$
Solve for x
Divide both sides by $-3$ to isolate $x$ .
$x = -\ \frac{2}{3}$

Answer:

x = -\frac{2}{3}

Check: Substitute

x = -\frac{2}{3}

:

\left(\frac{1}{27}\right)^{-2/3} = 27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9

. ✓

Common mistakes and how to avoid them

Multiplying the bases instead of using the power-of-a-power law. For example, writing

(2^3)^2 = 2^5

instead of

2^6

.

Correction: When a power is raised to another exponent, multiply the exponents:

(2^3)^2 = 2^{3 \times 2} = 2^6

.

Forgetting the negative sign when converting a fraction like

\frac{1}{9}

, and writing

\frac{1}{9} = 3^2

instead of

3^{-2}

.

Correction: Reciprocals always produce a negative exponent:

\frac{1}{9} = 9^{-1} = (3^2)^{-1} = 3^{-2}

.

Equating exponents when the bases on both sides are not actually the same after rewriting.

Correction: Check that every term on both sides uses the identical base before setting exponents equal.

Choosing a base that does not work for all terms — for example, trying base

6

for an equation involving

4

and

8

.

Correction: Choose the smallest positive integer base that all terms can be expressed as an exact integer power of — usually

2

,

3

, or

5

.

Skipping the substitution check and missing a sign error in the exponent.

Correction: Always substitute the answer back into the original equation to confirm both sides are equal.

Lesson summary

To rewrite $b^n$ in a new base $c$ , find $k$ so that $b = c^k$ , then apply the power-of-a-power law: $b^n = c^{kn}$ .
Two exponential equations are equivalent if they describe the same relationship and share the same solution, just written with different bases.
To connect or solve an exponential equation, rewrite all terms with a single common base and then equate the exponents.
The rule $c^A = c^B \Rightarrow A = B$ is valid only when $c > 0$ and $c \neq 1$ .
Fractional bases like $\frac{1}{b^k}$ are rewritten as $b^{-k}$ before applying the power-of-a-power law.
Always verify your solution by substituting it back into the original equation.

Check your understanding

Question 1

Which of the following is equivalent to

8^{2x}

, written with base

2

?

$2^{2x}$
$2^{5x}$
$2^{6x}$
$2^{8x}$

Show answer and explanation

2^{6x}

Since

8 = 2^3

, we get

8^{2x} = (2^3)^{2x} = 2^{3 \times 2x} = 2^{6x}

. The exponents are multiplied, not added.

Question 2

What is the value of

x

in the equation

9^x = 27

?

$x = 2$
$x = \frac{3}{2}$
$x = 3$
$x = \frac{2}{3}$

Show answer and explanation

x = \frac{3}{2}

Write both sides in base

3

:

9 = 3^2

and

27 = 3^3

, so

(3^2)^x = 3^3

gives

3^{2x} = 3^3

, meaning

2x = 3

and

x = \frac{3}{2}

. Check:

9^{3/2} = (\sqrt{9})^3 = 3^3 = 27

. ✓

Question 3

The equation

\left(\frac{1}{4}\right)^x = 8

rewritten entirely in base

2

is which of the following?

$2^{-2x} = 2^3$
$2^{2x} = 2^3$
$2^{-4x} = 2^3$
$2^{-2x} = 2^4$

Show answer and explanation

2^{-2x} = 2^3

Since

\frac{1}{4} = 2^{-2}

and

8 = 2^3

, substituting gives

(2^{-2})^x = 2^3

, which simplifies to

2^{-2x} = 2^3

.

Question 4

For the rule that lets you conclude equal exponents from equal powers to be valid, which condition on the base is required?

The base must equal $1$ .
The base can be any real number.
The base must be positive and not equal to $1$ .
The base must be an integer greater than $2$ .

Show answer and explanation

The base must be positive and not equal to

1

.

The rule

c^A = c^B \Rightarrow A = B

requires

c > 0

and

c \neq 1

. If

c = 1

, then

1^A = 1^B = 1

regardless of

A

and

B

, so the exponents could be anything — you cannot compare them.

Key terms

Base: The number that is repeatedly multiplied in a power expression. In $3^4$ , the base is $3$ .
Exponent: The number that tells you how many times the base is multiplied by itself. In $3^4$ , the exponent is $4$ .
Power: An expression of the form $b^n$ , which represents the repeated multiplication of the base $b$ a total of $n$ times.
Power-of-a-Power Law: The rule $(b^m)^n = b^{mn}$ : when a power is raised to another exponent, multiply the two exponents.
Equivalent Equations: Two equations that have exactly the same solution set, even if they look different on the surface.
Common Base: A single base that can be used to rewrite all terms in an equation, making it possible to compare exponents directly.
Negative Exponent: An exponent that is less than zero. The rule $b^{-n} = \frac{1}{b^n}$ converts between negative exponents and fractions.

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