Learn9MA0Paper 2 · Pure Mathematics 2Exponentials and logarithmsExponential equations and inequalities

Exponentials and logarithms · Subtopic 3 of 5

Exponential Equations and Inequalities

Bringing an unknown power down with logarithms: solving $a^x=b$ , equations with $x$ in two powers, disguised quadratics, and inequalities like "the smallest $n$ such that $0.8^n<0.1$ ".

Cambridge 9709 Paper 2Cambridge 9709 Paper 3Edexcel YMA01 WMA12Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1≈ 40 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve $a^x=b$ and $a^{px+q}=b$ by taking logarithms.
Solve equations with the unknown in two different powers, such as $2^{x+1}=5^x$ .
Spot and solve disguised quadratics such as $3^{2x}-4(3^x)+3=0$ .
Solve exponential inequalities, taking care when dividing by a negative logarithm.

Pause and think

$2^x=10$ . You know $2^3=8$ and $2^4=16$ . Between which two whole numbers is $x$ ? How could you find it exactly?

Section 2 of 14

Core idea

Start hereWhen will it double?

Money grows at $5\%$ a year: after $n$ years, $1.05^n$ times as much. When does it double? $1.05^n=2$ . The unknown is in the power, where ordinary algebra can't reach. Logarithms bring it down.

The big idea

To solve $a^x=b$ , take logs of both sides: $x\ln a=\ln b$ , so $x=\dfrac{\ln b}{\ln a}$ . Equations in $a^{2x}$ and $a^x$ are hidden quadratics: substitute $u=a^x$ .

1. Taking logs

3ˣ = 20 where the curve meets the line

3^2=9

and

3^3=27

, so the answer is between

2

and

3

, nearer

3

. A quick sense-check before using the calculator.

Worked exampleDoubling: $1.05^n=2$

Solve.

Goal: $n$ .
Strategy: Logs of both sides; power law.

$\ln(1.05^n)=\ln2$
Take $\ln$ of both sides (both positive).
$n\ln1.05=\ln2$
Power law.
$n=\frac{\ln2}{\ln1.05}=14.2$
Divide.

AnswerAbout

14.2

years (so 15 whole years).

✓ Check

1.05^{14.2}=2.00

✓.

Worked example $5^{2x-1}=3^{x+2}$

Solve to 3 s.f.

Goal: $x$ .
Strategy: Logs, expand, collect $x$ terms.

$(2x-1)\ln5=(x+2)\ln3$
Take logs; power law.
$2x\ln5-\ln5=x\ln3+2\ln3$
Expand.
$x(2\ln5-\ln3)=2\ln3+\ln5$
Collect $x$ terms on one side.
$x=\frac{2\ln3+\ln5}{2\ln5-\ln3}=\frac{3.807}{2.120}=1.80$
Divide.

Answer

x=1.80

✓ CheckWith

x=1.796

5^{2.592}\approx64.8

and

3^{3.796}\approx64.7

✓ (equal up to rounding).

2. Hidden quadratics

Worked example $2^{2x}-5(2^x)+4=0$

Solve.

Goal: All $x$ .
Strategy: $2^{2x}=(2^x)^2$ ; let $u=2^x$ .

$u^2-5u+4=0\Rightarrow(u-1)(u-4)=0$
Substitute; factorise.
$2^x=1\Rightarrow x=0;\quad2^x=4\Rightarrow x=2$
Undo.

Answer

x=0

2

✓ Check

x=2

16-20+4=0

✓.

Worked example $e^x-6e^{-x}=1$

Solve.

Goal: $x$ .
Strategy: Multiply by $e^x$ to remove the negative power.

$e^{2x}-6=e^x$
$e^{-x}\cdot e^x=1$ .
$u^2-u-6=0\Rightarrow(u-3)(u+2)=0$
$u=e^x$ .
$e^x=3\Rightarrow x=\ln3;\quad e^x=-2\ \text{impossible}$
$e^x>0$ always.

Answer

x=\ln3

✓ Check

3-\frac63=1

✓.

3. Inequalities

Worked exampleSmallest $n$ with $0.8^n<0.01$

Find it.

Goal: Whole $n$ .
Strategy: Logs; dividing by a negative log flips the sign.

$n\ln0.8<\ln0.01$
Take logs.
$n>\frac{\ln0.01}{\ln0.8}=20.6$
$\ln0.8<0$ : flip.

Answer

n=21

✓ Check

0.8^{20}=0.0115

0.8^{21}=0.0092

✓.

Common misconceptionForgetting to flip

\ln

of a number between

0

and

1

is negative.

Pause and think

Solve $e^{3x}=7$ .

Summary

Logs bring powers down.
$u=a^x$ for hidden quadratics; reject $a^x\le0$ .
Flip when dividing by a negative log.

Exponential equation: Unknown in the exponent.

HistoryDating the past. In 1949 Willard Libby showed that the amount of carbon-14 in dead material falls exponentially, halving about every

5730

years. Solving an exponential equation then gives the age of bones, wood or cloth. He won the 1960 Nobel Prize in Chemistry for it.

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