YEAR 10 · EXPONENTIAL RELATIONSHIPS · 15 MIN

Exponential equations

Solve exponential equations by matching bases or using logarithms when a matching base is unavailable.

In this lesson

Rewrite powers with a common base
Check solutions in the original equation

Before you start

Index laws
Fractional powers

01 / LEARN

Understand the idea.

Match the base first

For $b > 0$ and $b \neq 1$ , equal powers $b^{u} = b^{v}$ have equal exponents $u = v$ . Try rewriting both sides using one base; the exponential equation then reduces to an ordinary equation in the exponents.

Isolate the power before comparing bases. For $3 (2^{x}) = 24$ , divide by $3$ first to obtain $2^{x} = 8$ . The outside coefficient multiplies the power and is not part of its exponent.

When there is no convenient shared base

When no convenient common base exists, take logarithms. From $b^{x} = c$ with $c > 0$ , the power law gives $x log b = log c$ , so $x = \frac{log c}{log b}$ using one consistent logarithm base.

Retain the exact logarithm ratio until final rounding, and check by substitution. A positive-base exponential is always positive: if isolating the power gives a zero or negative right-hand side, there is no real solution.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Common base

Solve $4^{x} = 8$ .

Worked solution

1
Rewrite both numbers with base two.
${(2^{2})}^{x} = 2^{3}$
Both $4$ and $8$ are powers of $2$ .
2
Simplify and equate the exponents.
$2^{2 x} = 2^{3} ⟹ 2 x = 3$
The base- $2$ exponential is one-to-one.
3
Solve and check.
$x = \frac{3}{2}, 4^{\frac{3}{2}} = {(\sqrt{4})}^{3} = 8$
Dividing by $2$ gives the exponent and substitution verifies the original equation.

Answer

$x = 3 / 2$

EXAMPLE 02

Use a log ratio

Solve $3^{x} = 20$ to 3 decimal places.

Worked solution

1
Take logarithms on both sides.
$log (3^{x}) = log 20$
Both sides of the original equation are positive.
2
Use the power law and isolate the exponent.
$x log 3 = log 20 ⟹ x = \frac{log 20}{log 3}$
The same logarithm base is used throughout, and $log 3 \neq 0$ .
3
Evaluate to the requested precision.
$x = 2.726833 \dots \approx 2.727$
Round the logarithm ratio only at the final step.
4
Check the rounded result.
$3^{2.727} \approx 20.004$
The small difference from $20$ is consistent with rounding $x$ to three decimal places.

Answer

$x \approx 2.727$

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $5 (2^{x}) = 40$ .

Show answer & reasoning

$x = 3$

Divide by $5$ to get $2^{x} = 8 = 2^{3}$ . Equal powers of the same valid base give $x = 3$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Solve $2^{x} = 32$ .
2 marks
Solve $9^{x} = 27$ .
2 marks
Solve $5^{x - 1} = 125$ .
2 marks
Solve ${(\frac{1}{2})}^{x} = 8$ .
2 marks
Solve $2^{x} = 7$ to 3 decimal places.
3 marks
Explain whether $4^{x} + 2 = 1$ has a real solution.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $2^{x} = 32$ .
2 marks
$32 = 2^{5}$ .
$x = 5$
Solve $9^{x} = 27$ .
2 marks
$3^{2 x} = 3^{3}$ , so $2 x = 3$ .
$x = \frac{3}{2}$
Solve $5^{x - 1} = 125$ .
2 marks
$125 = 5^{3}$ , so $x - 1 = 3$ .
$x = 4$
Solve ${(\frac{1}{2})}^{x} = 8$ .
2 marks
${(\frac{1}{2})}^{x} = 2^{- x}$ and $8 = 2^{3}$ , so $- x = 3$ .
$x = - 3$
Solve $2^{x} = 7$ to 3 decimal places.
3 marks
$x = \frac{log 7}{log 2} = 2.807354 \dots$ .
$x \approx 2.807$
Explain whether $4^{x} + 2 = 1$ has a real solution.
3 marks
It would require $4^{x} = - 1$ , but $4^{x}$ is positive for every real $x$ .
No real solution.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

In this lesson

Understand the idea.

Match the base first

When there is no convenient shared base

See the method in action.

Common base

Rewrite both numbers with base two.

Simplify and equate the exponents.

Solve and check.

Use a log ratio

Take logarithms on both sides.

Use the power law and isolate the exponent.

Evaluate to the requested precision.

Check the rounded result.

Try it yourself.

Check your understanding

Keep practising