Logarithmic and exponential functions: past paper questions

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• Question 2May/June 2026, paper 31Structured5 marks

  The variables $x$ and $y$ satisfy the equation $y^3=A{\mathrm{e}}^{kx}$, where $A$ and $k$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(-1.32,5.09)$ and $(0.25,1.16)$. Find the values of $A$ and $k$.

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• Question 2Oct/Nov 2025, paper 31Structured5 marks

  Show that the equation ${\log }_4(2x+1)=2{\log }_4(3x-1)-2$ can be written as a quadratic equation in $x$.

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• Question 4Oct/Nov 2024, paper 32Structured3 marks

  Solve the equation $5^x=5^{x+2}-10$.

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• Question 3Oct/Nov 2023, paper 31Structured4 marks

  The variables $x$ and $y$ are related by the equation $y=a{b}^x$, where $a$ and $b$ are constants. The diagram shows the result of plotting $\ln y$ against $x$ for two pairs of values of $x$ and $y$. The coordinates of these points are $(1,3.7)$ and $(2.2,6.46)$.

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• Question 2Oct/Nov 2025, paper 32Structured4 marksHas a diagram

  Solve the equation $3\times 2^{x+1}=4\times 3^{2x-3}$.

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• Question 6Oct/Nov 2024, paper 32Structured4 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $ay=b^x$, where $a$ and $b$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(0.50,2.24)$ and $(3.40,8.27)$, as shown in the diagram. Find the values of $a$ and $b$.

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• Question 1Oct/Nov 2023, paper 33Structured4 marks

  Find the set of values of $x$ satisfying the inequality $\mathrm{\mid }{2}^{x+1}-2\mathrm{\mid }<0.5$, giving your answer to 3 significant figures.

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• Question 1May/June 2026, paper 33Structured4 marks

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

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• Question 3Oct/Nov 2025, paper 33Structured4 marks

  Solve the equation $2^{3x-4}=\frac{3}{5^x}$.

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• Question 3Oct/Nov 2024, paper 33Structured5 marks

  The number of bacteria in a population, $P$, at time $t$ hours is modelled by the equation $P=a{\mathrm{e}}^{kt}$, where $a$ and $k$ are constants. The graph of $\ln P$ against $t$, shown in the diagram, has gradient $\frac{1}{20}$ and intersects the vertical axis at $(0,3)$.

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• Question 1May/June 2023, paper 31Structured3 marks

  Solve the equation $$3{\mathrm{e}}^{2x}-4{\mathrm{e}}^{-2x}=5.$$ Give the answer correct to 3 decimal places.

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• Question 2May/June 2026, paper 33Structured4 marks

  Find the set of values of $x$ satisfying the inequality $\mathrm{\mid }{3}^{2x-1}-4\mathrm{\mid }>2$.

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• Question 2May/June 2025, paper 31Structured3 marks

  It is given that $2\ln p+\ln (p-1)-\frac{1}{2}\ln (q+1)=3$. Find $q$ in terms of $p$.

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• Question 2May/June 2024, paper 31Structured4 marks

  Solve the equation $\ln (x-5)=7-\ln x$.

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• Question 2May/June 2023, paper 32Structured3 marks

  Solve the equation $\ln (2x^2-3)=2\ln x-\ln 2$, giving your answer in an exact form.

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• Question 3May/June 2024, paper 31Structured4 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $a^y=bx$, where $a$ and $b$ are constants. The graph of $y$ against $\ln x$ is a straight line passing through the points $(0.336,1.00)$ and $(1.31,1.50)$, as shown in the diagram. Find the values of $a$ and $b$.

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• Question 1May/June 2023, paper 33Structured4 marks

  Solve the equation $\ln (x+5)=5+\ln x$.

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• Question 3May/June 2024, paper 32Structured5 marks

  The variables $x$ and $y$ satisfy the equation $a^{2y-1}=b^{x-y}$, where $a$ and $b$ are constants. Show that the graph of $y$ against $x$ is a straight line.

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• Question 1May/June 2024, paper 33Structured4 marks

  Solve the equation $8^{3-6x}=4\times 5^{-2x}$.

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• Question 4May/June 2024, paper 33Structured4 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $ky={\mathrm{e}}^{cx}$, where $k$ and $c$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(2.80,0.372)$ and $(5.10,2.21)$, as shown in the diagram. Find the values of $k$ and $c$.

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