Learn9709Paper 2 · Pure Mathematics 2Logarithmic and exponential functionsLogarithmic equations

Logarithmic and exponential functions · Subtopic 4 of 5

Logarithmic Equations

Solving equations with the unknown inside a logarithm: combine with the log laws, remove the logs by raising to a power, and always check that the answers are in the domain.

Cambridge 9709 Paper 2Cambridge 9709 Paper 3Edexcel YMA01 WMA12Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve $\log_ax=b$ by rewriting as $x=a^b$ .
Combine logs into a single log on each side, then remove them.
Solve the resulting linear or quadratic equation.
Reject any solution that would need the log of zero or a negative number.

Pause and think

$\lg x=2$ . Without a calculator, what is $x$ ?

Section 2 of 14

Core idea

Start hereUndoing a log

If $\log_2(\text{something})=4$ , the something is $2^4=16$ . Log equations are solved by combining everything into one log, then "un-logging". The catch: logs only accept positive inputs, so some algebraic answers must be thrown away.

The big idea

Combine logs with the laws into a single log, convert to an exponential with $\log_ab=c\iff b=a^c$ , solve, and check every log's argument is positive.

1. The domain restriction

Logs only exist for positive inputs

Nothing to the left of the

y

-axis:

\log_2x

is undefined for

x\le0

. That is why some algebraic answers must be rejected.

2. Examples

Worked example $\log_2(x+3)+\log_2(x-3)=4$

Solve.

Goal: Valid $x$ .
Strategy: Combine, un-log, solve, check.

$\log_2[(x+3)(x-3)]=4$
Product law.
$x^2-9=2^4=16$
Un-log.
$x^2=25\Rightarrow x=\pm5$
Solve.
$x=-5$ : $\log_2(-2)$ undefined. Reject.

Answer

x=5

✓ Check

\log_28+\log_22=3+1=4

✓.

Worked example $2\log_3x-\log_3(x+4)=1$

Solve.

Goal: $x$ .
Strategy: Power law, quotient law, un-log.

$\log_3\frac{x^2}{x+4}=1$
Combine.
$x^2=3(x+4)\Rightarrow x^2-3x-12=0$
Un-log; rearrange.
$x=\frac{3\pm\sqrt{57}}2$
Formula.

Answer

x=\frac{3+\sqrt{57}}2\approx5.27

(the negative root fails

x>0

✓ Check

x^2=27.8

x+4=9.27

, ratio

3.0

✓.

Worked exampleDifferent bases: $\log_2x=\log_4(x+6)$

Solve.

Goal: $x$ .
Strategy: Change base so both are base 2.

$\log_4(x+6)=\frac{\log_2(x+6)}{\log_24}=\frac{\log_2(x+6)}2$
Change of base.
$2\log_2x=\log_2(x+6)\Rightarrow x^2=x+6$
Multiply by 2; power law; equate.
$(x-3)(x+2)=0$
Solve; reject $-2$ .

Answer

x=3

✓ Check

\log_23=1.585

;

\log_49=1.585

✓.

Common misconception $\log(x+3)+\log(x-3)\neq\log(2x)$ The product law gives

\log[(x+3)(x-3)]

: multiply the insides, don't add them.

Pause and think

Solve $\ln x+\ln(x+2)=\ln8$ .

Summary

One log, then un-log.
Reject non-positive arguments.

Argument: The input of a log.

HistoryCommon logarithms. Henry Briggs visited John Napier in 1615 and suggested logarithms to base

10

. Base-10 ("common") logarithms, now written

\lg

, were printed in tables used by scientists, navigators and engineers for over three centuries.

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