Solving exponential and log equations: OCR A‑Level Maths questions

OCR A-Level Maths past-paper questions on 1.06.3 Solving exponential and log equations, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 23 questions on Solving exponential and log equations, as a list

June 2025, Paper 2, Q2 (9 marks): In this question you must show detailed reasoning. Solve the following equations. 2 3 (a) ( x 5 ) 8 [3] - = 2 3 = y (b) e 2 [2] 4 2 (c) x 3 x 4 0 [4] - - =
June 2025, Paper 3, Q3 (7 marks): (a) Sketch the graph of y x [2] 2 5 on the grid provided in the Printed Answer Booklet. = - (b) Solve the inequality x [2] 2 5 1 . - 1 (c) In this question you
June 2025, AS Paper 1, Q1 (8 marks): In this question you must show detailed reasoning. Solve the following equations. 2 - (a) x 11 = 5 [2] 6 3 (b) y 7 y 8 0 [4] + - = 10 = 3 z (c) 2 - 16 [2]
June 2025, AS Paper 1, Q6 (7 marks): In this question you must show detailed reasoning. Solve the following equations. (a) log ( x 32 ) log ( 2 x ) log 2 2 [4] + - + = 3 3 3 c m 1 3 (b) log – [3] =
June 2025, AS Paper 2, Q8 (6 marks): x y x 2 y 8 The curves e 2 e = and 2 1 e 3 e 4 1 intersect at the point P. - + = 2 + y y (a) Show that the y-coordinate of P satisfies the equation 3 e 4 e 39 0
June 2024, Paper 1, Q6 (8 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( )x f is defined by ( )x f 4 x 25 x 58 x 16 . = - - + 1 (a) Show that x is a root o
June 2024, AS Paper 1, Q1 (3 marks): In this question you must show detailed reasoning. ` j ` j 3 2 Solve the equation ln x x , where x 2 . [3] ln 5 ln 2 0 - =
June 2024, AS Paper 2, Q5 (7 marks): e x 3 5 At the point P on the curve y 21 x - the gradient of the tangent is 3. 8 = - (a) Determine the x-coordinate of P. Give your answer in the form ln a, whe
June 2023, Paper 1, Q2 (8 marks): 1 a 2 (a) (i) Show that can be written in the form b , where a, b and c are + cx 3 2 x 3 2 x + - + constants to be determined. [2] 1 1 (ii) Hence solve the equa
June 2023, Paper 3, Q1 (3 marks): Using logarithms, solve the equation 1 = x x 2 4 + 5 , giving your answer correct to 3 significant figures. [3]
June 2023, AS Paper 1, Q4 (4 marks): x 4 A curve has equation y e . = (a) Determine the value of x when y 10 . [2] = (b) Determine the gradient of the tangent to the curve at the point where x 2 .
June 2023, AS Paper 2, Q5 (5 marks): y 2 x 3 2 y + = 2 x 3 y = x O + . The graph of y 2 x 3 2 x 3 2 2 x 3 The diagram shows the graphs of y and y can be = = = 2 x 3 2 transformed to the graph of y
June 2022, Paper 1, Q5 (8 marks): 2x 2x 4 5 (a) The graph of y can be transformed to the graph of y + either by a translation or = = by a stretch. (i) Give full details of the translation. [2] (
June 2022, Paper 2, Q4 (5 marks): An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and
June 2022, AS Paper 2, Q6 (12 marks): y C x O A B The shape ABC shown in the diagram is a student’s design for the sail of a small boat. a k 3 The curve AC has equation y 2 log x and the curve BC ha
November 2021, Paper 1, Q4 (7 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( ) f x is defined by ( ) f x 2 x 3 x 11 x + . 6 = - - (a) Use the factor theorem to
November 2021, Paper 1, Q9 (10 marks): A particle moves in the x-y plane so that at time t seconds, where t 0 , its coordinates are given by H - . 2 t t 3 t x e 4 e 3 , y 2 e = - + = (a) Explain why
November 2020, AS Paper 1, Q6 (13 marks): In this question you must show detailed reasoning. 2 (a) Solve the inequality x x 6 0 , giving your answer in set notation. [4] 2 + - 3 3 (b) Solve the equation
November 2020, AS Paper 2, Q4 (9 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial x 6 kx 57 x 20 is denoted by f (x). It is given that ( 2 x 1 ) is a factor of + + -
June 2019, Paper 1, Q7 (8 marks): In this question you must show detailed reasoning. n A sequence u , u , u f is defined by u 25 0 6 . . = # 1 2 3 n N 3 / / 4 Use an algebraic method to find the
June 2019, AS Paper 1, Q7 (11 marks): ekx 7 (a) Write down an expression for the gradient of the curve y . [1] = 1 e x (b) The line L is a tangent to the curve y at the point where x 2 . = = 2 Show
June 2019, AS Paper 2, Q8 (8 marks): (a) Show that the equation x kx where k is a constant, can be expressed in 2 log log ( 1 ) 3 , = - + 2 2 2 - x kx [4] the form 8 8 0 . + = (b) Given that the eq
June 2018, AS Paper 2, Q6 (9 marks): A pan of water is heated until it reaches 100 °C. Once the water reaches 100 °C, the heat is switched off and the temperature T °C of the water decreases. The t

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