STEM-TutorLearn • Practice • ExcelBook Free DemoFree demo
AP CoursesGCSEInstructorsTestimonials
Guides
Past papers
ContactBook Free Demo

Exponentials and logarithms practice problems A level Maths, Topic 4

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 exam-style question

The laws of logarithms, solving exponential equations, e and natural logarithms, and exponential growth and decay models. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Exam question practice
  5. Common mistakes
  6. Keep going

Key equations for exponentials and logarithms

log a + log b = log(ab)product law
log an = n log apower law
ax = b ⇔ x = loga bdefinition
y = Aektexponential growth (k > 0) or decay (k < 0)

A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
Evaluate log2 32.

Hint

2 to what power gives 32?

Worked solution
  1. 25 = 32.
  2. log2 32 = 5.
Q2Warm-up
Solve 3x = 20, to 3 significant figures.

Hint

Take logs of both sides.

Worked solution
  1. x = ln 20/ln 3.
  2. x = 2.73.
Q3Warm-up
Which is equal to log a + log b?

Hint

The product law.

Worked solution
  1. Adding logs multiplies the arguments: log a + log b = log(ab).
  2. The answer is B.

Exam-style questions

Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Solve ln x = 2, to 3 significant figures.

Hint

Undo ln with e.

Worked solution
  1. x = e2.
  2. x = 7.39.
Q5Exam-style
Solve log3 x + log3 9 = 4.

Hint

log3 9 = 2.

Worked solution
  1. log3 x = 4 − 2 = 2.
  2. x = 32 = 9.
Q6Exam-style
A population is modelled by N = 500e−0.2t. Find N when t = 5, to the nearest whole number.

Hint

Substitute t = 5.

Worked solution
  1. N = 500e−1 = 500 × 0.3679.
  2. N = 184.
Q7Exam-style
For the same model, find the time taken for N to halve, to 3 significant figures.

Hint

Set e−0.2t = ½.

Worked solution
  1. −0.2t = ln ½, so t = ln 2/0.2.
  2. t = 3.47.
Q8Challenge
Solve 22x − 6(2x) + 8 = 0. Give the larger solution.

Hint

Let y = 2x, so 22x = y2.

Worked solution
  1. y2 − 6y + 8 = 0 gives y = 2 or y = 4.
  2. 2x = 2 gives x = 1; 2x = 4 gives x = 2.
  3. Larger solution: 2.

Exam question practice

One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.

Exam-style question style · 8 points · about 8 minutes

The value of a machine, £V, t years after it is bought is modelled by V = 20 000e−kt. After 2 years it is worth £15 000.

  1. (a) Show that k = ½ ln(4/3).
  2. (b) Find the value after 5 years, to the nearest pound.
  3. (c) Find when the value first falls below £5000, to 1 decimal place.
  4. (d) Give one limitation of this model.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

15 000 = 20 000e−2k, so e−2k = 3/4 and −2k = ln(3/4), giving k = ½ ln(4/3).

Part (b)

k = 0.1438. V = 20 000e−0.719 = £9743.

Part (c)

e−kt < 1/4, so t > ln 4/k = 9.6 years.

Part (d)

It predicts the value never reaches zero, or it ignores factors such as condition and demand.

Your mark: 0 of 8

Common mistakes in exponentials and logarithms

Keep going

Revise the ideas behind these questions:

Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units