A-Level Maths: Exponentials and Logarithms

Exponentials and logarithms connect A-Level algebra to the real-world modelling questions that anchor every paper — growth, decay, and the log-linearisation of data. This quiz covers the section end to end.

The log-law questions build fluency in both directions: recognising which expressions equal ln 12 (2ln 2 + ln 3 and ln 4 + ln 3, against near-misses that multiply to 18 or 32), and computing log_a(x²y) = 8 from given values by the power and product laws.

The e-and-ln questions test the inverse relationship precisely: reflections in y = x, the range of eˣ, the domain of ln x, and the derivative of eˣ — with a distractor built from the power-rule misuse xe^(x−1) that examiners see every year. A simplification question combines both directions in one expression: e^(3ln 2) − ln(e⁴) = 8 − 4.

Equation-solving covers the disguised quadratic e^(2x) − 5eˣ + 6 = 0, solved by the substitution u = eˣ and yielding two log answers, and the log equation ln(x+3) + ln(x−1) = ln 5, where the quadratic's negative root must be rejected against the domain — the checking step that separates full marks from lost ones.

The modelling questions close the set: linearising y = abˣ by taking logs (ln y against x gives a straight line), the parallel case y = axⁿ (log y against log x), reading initial value and decay factor from M = 100(0.8)ᵗ, and interpreting the transformation from ln x to ln(x+2) − 3 as a translation.

Every explanation traces the law being applied, making the set a complete rehearsal for the "use logarithms to..." questions that appear on every specification.

Apply the laws of logarithms fluently in both directions
Use the inverse relationship between e^x and ln x, including their graphs
Solve exponential equations via substitution and log equations with domain checks
Linearise y = ab^x and y = ax^n relationships using logs
Interpret exponential growth and decay models in context

Topic scope follows section F (Exponentials and logarithms) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: log laws, exponential equations, e^x and ln x, reduction to linear form, and growth and decay modelling.

Sample question

Which of the following expressions are equivalent to $\ln(12)$?

$2\ln(2) + \ln(3)$
$\ln(6) + \ln(3)$
$\ln(4) + \ln(3)$
$\ln(8) + \ln(4)$

See the answer

$2\ln(2) + \ln(3)$, $\ln(4) + \ln(3)$

Using log laws: $\ln(12) = \ln(2^2 \cdot 3) = 2\ln(2) + \ln(3)$, and $\ln(12) = \ln(4 \cdot 3) = \ln(4) + \ln(3)$. By contrast, $\ln(6) + \ln(3) = \ln(18)$ and $\ln(8) + \ln(4) = \ln(32)$.

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