Exponentials & logarithms

Edexcel Pure 6

A-level Maths (9MA0) · exam-style practice, examiner-report intelligence and the tools that drill it.

Board and spec code confirmed against Edexcel 9MA0 · registry checked 2026-07-11How this checking works

Edexcel 9MA0 revision notes Edexcel 9MA0 past papers

The topic on one screen

The three log laws do all the work: $\log a + \log b = \log(ab)$ , $\log a - \log b = \log(a/b)$ , $k\log a = \log(a^k)$ .
$\log_a(x) = y$ means $a^y = x$ . When stuck, convert to index form.
Solving $a^x = b$ : take logs of both sides, $x = \dfrac{\log b}{\log a}$ — exact form first, decimal only if asked.
$e$ and $\ln$ are inverses: $e^{\ln x} = x$ and $\ln(e^x) = x$ . $\ln$ appears the moment calculus is involved.
Log-linear modelling: $V = ab^t$ becomes $\log V = \log a + t\log b$ — a straight line in $\log V$ against $t$ . Intercept gives $a$ , gradient gives $b$ .
A quadratic in disguise: $3^{2x}$ is $(3^x)^2$ — substitute $y = 3^x$ and factorise.

Where students actually lose marks

Log-manipulation marks require simplest form — an answer left as a sum of logs when a single log is demanded, or with un-processed coefficients, loses the final accuracy mark.

June 2023 Paper 1 mark scheme (Q06)

In the exponential-model question the marks come from linking the printed graph to the algebra: intercept $= \log a$ and gradient $= \log b$ . Students who quote $a$ and $b$ without that link lose the reasoning marks.

June 2023 Paper 1 mark scheme (Q11)

Where an exact answer is asked for, working with surds/logs must stay exact throughout — rounding mid-working forfeits the accuracy marks (general guidance printed in every scheme).

June 2023 Paper 1 mark scheme (general guidance)

Try it — exam-style

Medium

ORIGINAL · after June 2023 Paper 1 Q06

Given

a = \log_2 x

and

b = \log_2(x + 8)

, express in terms of

a

and/or

b

: (i)

\log_2(x^3)

, (ii)

\log_2(x^2 + 8x)

(3)

(Total for Question 1 is 3 marks)

Hard

ORIGINAL · after June 2023 Paper 1 Q11

The value

V

of a car,

t

years after purchase, is modelled by

V = ab^t

. The graph of

\log_{10} V

against

t

is a straight line through

(0, 4)

and

(20, 3.5)

. Find

a

and

b

(4)

(Total for Question 2 is 4 marks)

Hard

ORIGINAL

Solve

3^{2x} - 5(3^x) + 4 = 0

, giving exact answers.

(4)

(Total for Question 3 is 4 marks)

Easy

ORIGINAL

Solve

\ln(2x + 3) = 2

, giving your answer exactly.

(2)

(Total for Question 4 is 2 marks)

Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.

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Drill it properly

Formulae to Memorise

Log laws and exact values, on one page with a recall drill.

Graph Transformations

What e^x and ln x look like under translations and stretches.

Maths flashcards

Log laws and model-fitting prompts, on a review schedule.

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