Exponentials & logarithms
A-level Maths (9MA0) · exam-style practice, examiner-report intelligence and the tools that drill it.
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(ak).
- log_a(x) = y means ay = x. When stuck, convert to index form.
- Solving ax = b: take logs of both sides, x = log b / log a — exact form first, decimal only if asked.
- e and ln are inverses: e^(ln x) = x and ln(ex) = x. ln appears the moment calculus is involved.
- Log-linear modelling: V = abt 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 (3x)2 — substitute y = 3x 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
Given a = log2 x and b = log2 (x + 8), express in terms of a and/or b: (i) log2 (x3), (ii) log2 (x2 + 8x).
The value V of a car, t years after purchase, is modelled by V = abt. The graph of log10 V against t is a straight line through (0, 4) and (20, 3.5). Find a and b.
Solve 3^(2x) - 5(3x) + 4 = 0, giving exact answers.
Solve ln(2x + 3) = 2, giving your answer exactly.
Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.
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