A Level Maths: Pure 3 chapter 2: Logarithmic and exponential functions MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

29 exam-style questions on Logarithmic and exponential functions, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

29 questions · 1 topics · 10 per test

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What chapter 2 covers

Ten Logarithmic and exponential functions questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. Given that 2 log5 y − log5 x = 1, where x and y are positive, express y in terms of x.

  1. A
    y = 5x/2
  2. B
    y = √(x + 5)
  3. C
    y = 5√x
  4. D
    y = √(5x)
Show answer

Answer: D. log5(y2/x) = 1, so y2/x = 5 and y = √(5x). Writing 2 log5 y as log5 2y gives the wrong 5x/2.

2. Solve the equation ln(2x − 1) = 3, giving the exact answer.

  1. A
    x = (e3 − 1)/2
  2. B
    x = e3/2 + 1
  3. C
    x = (e3 + 1)/2
  4. D
    x = (ln 3 + 1)/2
Show answer

Answer: C. ln and e are inverse functions, so 2x − 1 = e3, 2x = e3 + 1 and x = (e3 + 1)/2 ≈ 10.5. Both terms must be divided by 2.

3. Solve the equation e2x = 7, giving the exact answer.

  1. A
    x = ln 3.5
  2. B
    x = ½ ln 7
  3. C
    x = 2 ln 7
  4. D
    x = ½e7
Show answer

Answer: B. Take ln of both sides: 2x = ln 7, so x = ½ ln 7 (≈ 0.973). Note ½ ln 7 = ln √7, not ln(7/2); and the inverse of ex is ln, so 2x = e7 is wrong.

4. Which statement about the graph of y = e−2x is correct?

  1. A
    It meets the y-axis at (0, 1) and y increases as x increases
  2. B
    It meets the y-axis at (0, 1) and y decreases as x increases
  3. C
    It meets the y-axis at (0, −2) and y decreases as x increases
  4. D
    It meets the x-axis at (1, 0) and y decreases as x increases
Show answer

Answer: B. At x = 0, y = e0 = 1. For y = ekx with k negative the curve decreases and approaches the x-axis as x → ∞ without ever meeting it.

5. Solve the equation 4x − 6(2x) + 8 = 0.

  1. A
    x = 1 or x = 2
  2. B
    x = 2 or x = 4
  3. C
    x = 0.693 or x = 1.39
  4. D
    x = 1 only
Show answer

Answer: A. 4x = (2x)2, so with u = 2x: u2 − 6u + 8 = (u − 2)(u − 4) = 0. Then 2x = 2 or 4, giving x = 1 or 2. The numbers 2 and 4 are values of u, not of x.

6. The graph of ln y against ln x is a straight line with gradient −0.5 that meets the ln y axis at 1.6. Express y in terms of x.

  1. A
    y = 4.95x−0.5
  2. B
    y = 1.6x−0.5
  3. C
    y = 4.95e−0.5x
  4. D
    y = −0.5x1.6
Show answer

Answer: A. ln y = 1.6 − 0.5 ln x, so y = e1.6 × x−0.5 = 4.95x−0.5. An exponential in x would come from a graph of ln y against x, not ln x.

7. A quantity P is modelled by P = 500e0.04t, where t is the time in years. Find the value of t for which P = 1500, correct to 3 significant figures.

  1. A
    t = 75.0
  2. B
    t = 27.5
  3. C
    t = 183
  4. D
    t = 0.0439
Show answer

Answer: B. e0.04t = 3, so 0.04t = ln 3 and t = ln 3 ÷ 0.04 = 27.5. Taking ln 1500 without first dividing by 500 gives 183.

8. Which statement about the graph of y = ln x is correct?

  1. A
    It crosses the x-axis at (1, 0) and the y-axis is an asymptote
  2. B
    It crosses the y-axis at (0, 1) and the x-axis is an asymptote
  3. C
    It passes through the origin and exists for all real x
  4. D
    It crosses the x-axis at (e, 0) and the y-axis is an asymptote
Show answer

Answer: A. ln 1 = 0, so the curve crosses the x-axis at (1, 0). ln x exists only for x > 0 and ln x → −∞ as x → 0 from above, so the y-axis is an asymptote. The description with (0, 1) fits y = ex, the reflection of y = ln x in y = x.

9. Solve the inequality 0.6x < 0.05, giving the boundary value correct to 3 significant figures.

  1. A
    x < 5.86
  2. B
    x > 0.0833
  3. C
    x > 5.86
  4. D
    x < −5.86
Show answer

Answer: C. x ln 0.6 < ln 0.05. Because ln 0.6 is negative, dividing by it reverses the inequality: x > ln 0.05 ÷ ln 0.6 = 5.86. A check: 0.66 ≈ 0.047, which is below 0.05.

10. Find the smallest integer n for which 1.08n > 5.

  1. A
    n = 20
  2. B
    n = 63
  3. C
    n = 21
  4. D
    n = 2
Show answer

Answer: C. n ln 1.08 > ln 5, so n > ln 5 ÷ ln 1.08 = 20.9. The smallest integer is therefore 21; rounding 20.9 down to 20 gives a value for which 1.0820 ≈ 4.66, still below 5.

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Ways to practise this chapter

Logarithmic and exponential functions: questions students ask

How many Logarithmic and exponential functions MCQs are there?

29, each with the answer and a short explanation, across 1 topics: Logarithmic and exponential functions. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 chapter 2 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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