plumiExponentials and logarithms: AQA A‑Level Maths questions
Every AQA A-Level Maths exam question on F Exponentials and logarithms, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 1, question 3: Express log3 2x – log3 x as a single logarithm. Circle your answer. [1 mark] log3 1 2 log3 2 log3 x log3 2 x2](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun25-03-qp.webp)
![June 2025 Paper 1, question 7: It is given that 0 < a < 1 Sketch the graph with equation y = ax on the axes below. [2 marks] x O y](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun25-07-qp.webp)
![June 2025 Paper 2, question 4: Solve the equation 5x–1 = 20 Give your answer in an exact form. [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun25-04-qp.webp)
![June 2025 Paper 3, question 3: The function f is defined by f(x) = ex for x ∈ ℝ Identify which one of the following statements describes the function f Tick (✓) one box. [1 mark] Decreasing a](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun25-03-qp.webp)
![June 2025 Paper 3, question 4: (a) State the value of loga a2 where a > 0 [1 mark] 4 (b) Find the value of loga √ a – loga 1 a2 where a > 0 [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun25-04-qp.webp)
![June 2024 Paper 1, question 2: The function f is defined by f (x) = ex + 1 for x ℝ Find an expression for f –1(x) Tick (✓) one box. [1 mark] f –1(x) = ln (x – 1) f –1(x) = ln (x) – 1 f –1(x)](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-02-qp.webp)
![June 2024 Paper 1, question 14: (a) The equation x3 = e6 – 2x has a single solution, x = α By considering a suitable change of sign, show that α lies between 0 and 4 [2 marks] 14 (b) Show that](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-14-qp.webp)
![June 2024 Paper 2, question 4: Use logarithms to solve the equation 5x–2 = 71570 Give your answer to two decimal places. [3 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun24-04-qp.webp)
![June 2024 Paper 3, question 3: One of the graphs shown below cannot have an equation of the form y = ax where a > 0 Identify this graph. Tick (✓) one box. [1 mark] y x O y x O](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun24-03-qp.webp)
![June 2024 Paper 3, question 4: A curve has equation y = x4 + 2x Find an expression for dy dx [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun24-04-qp.webp)
![June 2023 Paper 1, question 6: Show that the equation 2 log10 x = log10 4 + log10 (x + 8) has exactly one solution. Fully justify your answer. [5 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun23-06-qp.webp)
![June 2023 Paper 3, question 5: A curve has equation y = 3e2x Find the gradient of the curve at the point where y = 10 [3 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun23-05-qp.webp)
![June 2022 Paper 2, question 9: Given that log2 x3 − log2 y2 = 9 show that x = Ay p where A is an integer and p is a rational number. [4 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun22-09-qp.webp)
All 34 questions on Exponentials and logarithms, as a list
- June 2025, Paper 1, Q3 (1 mark): Express log3 2x – log3 x as a single logarithm. Circle your answer. [1 mark] log3 1 2 log3 2 log3 x log3 2 x2
- June 2025, Paper 1, Q7 (2 marks): It is given that 0 < a < 1 Sketch the graph with equation y = ax on the axes below. [2 marks] x O y
- June 2025, Paper 1, Q10 (12 marks): A researcher working for a frozen-food manufacturer uses the formula θ = 21 – Ae–kt to model the temperature of a dessert once it is taken out of a freezer. In
- June 2025, Paper 2, Q4 (2 marks): Solve the equation 5x–1 = 20 Give your answer in an exact form. [2 marks]
- June 2025, Paper 3, Q3 (1 mark): The function f is defined by f(x) = ex for x ∈ ℝ Identify which one of the following statements describes the function f Tick (✓) one box. [1 mark] Decreasing a
- June 2025, Paper 3, Q4 (3 marks): (a) State the value of loga a2 where a > 0 [1 mark] 4 (b) Find the value of loga √ a – loga 1 a2 where a > 0 [2 marks]
- June 2024, Paper 1, Q2 (1 mark): The function f is defined by f (x) = ex + 1 for x ℝ Find an expression for f –1(x) Tick (✓) one box. [1 mark] f –1(x) = ln (x – 1) f –1(x) = ln (x) – 1 f –1(x)
- June 2024, Paper 1, Q14 (10 marks): (a) The equation x3 = e6 – 2x has a single solution, x = α By considering a suitable change of sign, show that α lies between 0 and 4 [2 marks] 14 (b) Show that
- June 2024, Paper 2, Q4 (3 marks): Use logarithms to solve the equation 5x–2 = 71570 Give your answer to two decimal places. [3 marks]
- June 2024, Paper 2, Q8 (7 marks): A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula y = a + b log10 x where y is the median mass in
- June 2024, Paper 3, Q3 (1 mark): One of the graphs shown below cannot have an equation of the form y = ax where a > 0 Identify this graph. Tick (✓) one box. [1 mark] y x O y x O
- June 2024, Paper 3, Q4 (2 marks): A curve has equation y = x4 + 2x Find an expression for dy dx [2 marks]
- June 2024, Paper 3, Q8 (8 marks): The temperature θ °C of an oven t minutes after it is switched on can be modelled by the equation θ = 20(11 – 10e–kt) where k is a positive constant. Initially
- June 2023, Paper 1, Q3 (1 mark): The curve with equation y = ln x is transformed by a stretch parallel to the x-axis with scale factor 2 Find the equation of the transformed curve. Circle your
- June 2023, Paper 1, Q6 (5 marks): Show that the equation 2 log10 x = log10 4 + log10 (x + 8) has exactly one solution. Fully justify your answer. [5 marks]
- June 2023, Paper 2, Q6 (6 marks): Victoria, a market researcher, believes the average weekly value, £V million, of online grocery sales in the UK has grown exponentially since 2009. Victoria mod
- June 2023, Paper 3, Q5 (3 marks): A curve has equation y = 3e2x Find the gradient of the curve at the point where y = 10 [3 marks]
- June 2022, Paper 1, Q3 (1 mark): The curve y = log4 x is transformed by a stretch, scale factor 2, parallel to the y-axis. State the equation of the curve after it has been transformed. Circle
- June 2022, Paper 2, Q9 (4 marks): Given that log2 x3 − log2 y2 = 9 show that x = Ay p where A is an integer and p is a rational number. [4 marks]
- June 2022, Paper 2, Q10 (15 marks): A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects. Initially there are 25 dam
- June 2022, Paper 3, Q7 (7 marks): A planet takes T days to complete one orbit of the Sun. T is known to be related to the planet’s average distance d, in millions of kilometres, from the Sun. A
- November 2021, Paper 1, Q9 (15 marks): The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years. Year 1980 1985 1990 1995 200
- November 2021, Paper 2, Q6 (4 marks): Show that the solution of the equation 5x = 3x+4 can be written as x = ln 81 ln 5 − ln 3 Fully justify your answer. [4 marks]
- November 2020, Paper 2, Q1 (1 mark): Which one of these functions is decreasing for all real values of x? Circle your answer. [1 mark] f (x) = ex f (x) = −e1−x f (x) = −ex−1 f (x) = −e−x
- November 2020, Paper 3, Q5 (9 marks): The number of radioactive atoms, N, in a sample of a sodium isotope after time t hours can be modelled by N = N0e−kt where N0 is the initial number of radioacti
- November 2020, Paper 3, Q8 (12 marks): The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18 8 (a) Find the first term and c
- June 2019, Paper 1, Q1 (1 mark): Given that a > 0 , determine which of these expressions is not equivalent to the others. Circle your answer. [1 mark] −2 log10 1 a 2 log10 (a) log10 (a2) −4 log
- June 2019, Paper 2, Q8 (11 marks): Theresa bought a house on 2 January 1970 for £8000. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are
- June 2019, Paper 3, Q8 (12 marks): A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initia
- June 2018, Paper 1, Q2 (1 mark): The graph of y = 5x is transformed by a stretch in the y-direction, scale factor 5 State the equation of the transformed graph. Circle your answer. [1 mark] y =
- June 2018, Paper 1, Q4 (3 marks): The function f is defined by f (x) = e x−4, x 2 R Find f −1(x) and state its domain. [3 marks]
- June 2018, Paper 1, Q10 (8 marks): A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using m = m0e−kt where m0 milligrams is the initial mass of caff
- June 2018, Paper 1, Q11 (10 marks): The daily world production of oil can be modelled using V = 10 + 100 t 30 3 −50 t 30 4 where V is volume of oil in millions of barrels, and t is time in years s
- June 2018, Paper 3, Q7 (5 marks): (a) Given that loga y = 2 loga 7 + loga 4 + 1 2 , find y in terms of a. [4 marks] 7 (b) When asked to solve the equation 2 loga x = loga 9 − loga 4 a student gi
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