Logarithmic and exponential functions: past paper questions

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• Question 1May/June 2026, paper 21Structured4 marks

  The variables $x$ and $y$ satisfy the equation $3^{x+y}=2^{4x+1}$. Using natural logarithms, the equation can be transformed to the form $y=mx+c$. Determine the exact values of $m$ and $c$.

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• Question 2Oct/Nov 2025, paper 21Structured3 marks

  Solve the equation ${\mathrm{e}}^{2x}({\mathrm{e}}^{2x}-8)=48$.

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• Question 1Oct/Nov 2024, paper 21Structured5 marks

  The variables $x$ and $y$ satisfy the equation $a^{2y}={\mathrm{e}}^{3x+k}$, where $a$ and $k$ are constants. The graph of $y$ against $x$ is a straight line. Use logarithms to show that the gradient of the straight line is $\frac{3}{2\ln a}$.

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• Question 4Oct/Nov 2023, paper 21Structured7 marks

  Sketch, on the same diagram, the graphs of $y=\mathrm{\mid }3x-5\mathrm{\mid }$ and $y=2x+7$.

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• Question 3May/June 2026, paper 22Structured5 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $y=k{a}^{x+2}$, where $k$ and $a$ are positive constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(1.5,3.22)$ and $(4.0,4.69)$, as shown in the diagram.

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• Question 5Oct/Nov 2025, paper 21Structured7 marks

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-\frac{1}{2}x}-1$. The curve meets the axes at the points $A$ and $B$. The shaded region is bounded by the curve and the line segment $AB$.

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• Question 1Oct/Nov 2024, paper 22Structured3 marks

  Use logarithms to show that the equation $5^{8y}=6^{7x}$ can be expressed in the form $y=kx$.

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• Question 3Oct/Nov 2023, paper 22Structured5 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{-\frac{1}{2}x}$. The points on the curve with $x$-coordinates 0 and 2 are denoted by $A$ and $B$ respectively.

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• Question 1Oct/Nov 2025, paper 22Structured4 marks

  Solve the equation $\ln (3x+5)-\ln (x-2)=4$.

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• Question 4Oct/Nov 2023, paper 22Structured9 marks

  Sketch, on the same diagram, the graphs of $y=\mathrm{\mid }3-x\mathrm{\mid }$ and $y=9-2x$.

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• Question 3May/June 2024, paper 21Structured8 marks

  Sketch on the same diagram the graphs of $y=\mathrm{\mid }3x-8\mathrm{\mid }$ and $y=5-x$.

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• Question 3Oct/Nov 2023, paper 23Structured6 marks

  Find ${\int }_4^{10}\frac{4}{2x-5}\mathrm{d}x$, giving your answer in the form $\ln a$, where $a$ is an integer.

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• Question 2Oct/Nov 2025, paper 23Structured3 marks

  Solve the equation ${\mathrm{e}}^{2x}({\mathrm{e}}^{2x}-8)=48$.

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• Question 2May/June 2024, paper 22Structured4 marks

  Use logarithms to solve the equation $6^{2x-1}=5{\mathrm{e}}^{3x+2}$.

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• Question 3May/June 2024, paper 22Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-x}-{\mathrm{e}}^{2x}$. The curve crosses the $y$-axis at the point $A$ and the $x$-axis at the point $B$. The shaded region is bounded by the curve and the two axes.

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• Question 1May/June 2023, paper 21Structured4 marks

  Use logarithms to solve the equation ${12}^x=3^{2x+1}$.

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• Question 2May/June 2025, paper 21Structured6 marks

  Use logarithms to solve the inequality $4^x<0.05$.

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• Question 2May/June 2023, paper 22Structured5 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $y=A{\mathrm{e}}^{(A-B)x}$, where $A$ and $B$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(0.4,3.6)$ and $(2.9,14.1)$, as shown in the diagram.

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