A-level · Exponentials & logarithms

LogarithmsExam-style questions

A logarithm is the inverse of an exponential — it answers "what power?". Why logs exist (naming the solution of equations like 2^x = 100), converting between log and index form, evaluating logs, the special values, and natural logs (ln).

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2 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4

1
Find the value of $\log_8 32$ .
Level 3 · 2 marksTry it
2
A biologist is studying the growth of young trout in a fish hatchery.

The length, $L$ cm, of one of the trout is modelled by the equation

$L = 2 + 3\log_a(t + 1) \qquad\qquad 2 \leqslant t \leqslant 30$

where $t$ is the age of the trout in months and $a$ is a constant.

The length of this trout was $23$ cm when $t = 15$
(a) Show that $a = 1.486$ correct to 3 decimal places.

(3 marks)

In parts (b) and (c), use $a = 1.486$
(b) Make $t$ the subject of the equation of the model.

(3 marks)
(c) Find, according to the model, the age of the trout when its length is $26$ cm. Give your answer in months to 3 significant figures.

(2 marks)
Level 4 · 8 marksTry it