1.06g Equations with exponentials: solve a^x = b

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OCR MEI Paper 2 2022 June Q3
6 marks Moderate -0.8
  1. On the axes in the Printed Answer Booklet, sketch the curve with equation \(y = 3 \times 0.4^x\). [3]
  2. Given that \(3 \times 0.4^x = 0.8\), determine the value of \(x\) correct to 3 significant figures. [3]
WJEC Unit 1 2019 June Q10
13 marks Standard +0.3
  1. Solve the following simultaneous equations. $$3^{3x} \times 9^y = 27$$ $$2^{-3x} \times 8^{-y} = \frac{1}{64}$$ [6]
  2. Find the value of \(x\) satisfying the equation $$\log_a 3 + 2\log_a x - \log_a(x - 1) = \log_a(5x + 2).$$ [7]
WJEC Unit 1 2022 June Q10
3 marks Moderate -0.8
Showing all your working, solve the equation \(2^x = 53\). Give your answer correct to two decimal places. [3]
WJEC Unit 1 2023 June Q10
11 marks Moderate -0.3
Solve the following equations for values of \(x\).
  1. \(\ln(2x + 5) = 3\) [2]
  2. \(5^{2x+1} = 14\) [3]
  3. \(3\log_7(2x) - \log_7(8x^2) + \log_7 x = \log_3 81\) [6]
WJEC Unit 1 Specimen Q15
8 marks Moderate -0.8
The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = Ae^{kt}\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).
  1. Interpret the constant \(A\) in the context of the question. [1]
  2. Show that \(k = 0.047\), correct to three decimal places. [4]
  3. Find the size of the population when \(t = 20\). [3]
SPS SPS SM 2020 June Q12
8 marks Standard +0.3
\includegraphics{figure_6} **In this question you must show all stages of your working.** **Solutions relying on calculator technology are not acceptable.** Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2^{2x}$$ The point \(P\left(a, 96\sqrt{2}\right)\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3 \times 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is $$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]
SPS SPS SM 2020 October Q6
5 marks Moderate -0.8
  1. A student was asked to solve the equation \(2^{2x+4} - 9(2^x) = 0\). The student's attempt is written out below. $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ $$\text{Let } y = 2^x$$ $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$ Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2^{2x+4} - 9(2^x) = 0\), giving your answer in exact form. [3]
SPS SPS SM Pure 2021 May Q3
6 marks Standard +0.3
Solve the equation \(2^{4x-1} = 3^{5-2x}\), giving your answer in the form \(x = \frac{\log_{10} a}{\log_{10} b}\). [6]
SPS SPS SM 2022 October Q1
4 marks Easy -1.2
  1. Sketch the curve \(y = 3^{-x}\) [2]
  2. Solve the inequality \(3^{-x} < 27\) [2]
SPS SPS SM 2022 October Q4
8 marks Standard +0.3
  1. Find the positive value of \(x\) such that $$\log_x 64 = 2$$ [2]
  2. Solve for \(x\) $$\log_2(11 - 6x) = 2\log_2(x - 1) + 3$$ [6]
SPS SPS SM 2022 October Q6
6 marks Easy -1.2
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
  2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]
SPS SPS SM Pure 2022 June Q16
7 marks Standard +0.8
\includegraphics{figure_6} In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Figure 6 shows a sketch of part of the curve with equation $$y = 3x \cdot 2^{2x}.$$ The point \(P(a, 96\sqrt{2})\) lies on the curve.
  1. Find the exact value of \(a\). [3]
The curve with equation \(y = 3x \cdot 2^{2x}\) meets the curve with equation \(y = 6^{3-x}\) at the point \(Q\).
  1. Show that the \(x\) coordinate of \(Q\) is \(\frac{3 + 2\log_2 3}{3 + \log_2 3}\). [4]
SPS SPS SM 2021 November Q5
4 marks Moderate -0.3
  1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
  2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]
SPS SPS SM Pure 2023 June Q13
6 marks Moderate -0.8
A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N\) ppm (parts per million), is modelled by the equation $$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$ where \(t\) hours is the time after the treatment was applied. Use the equation of the model to answer parts (a) and (b).
  1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3] For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
  2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. [3]
SPS SPS FM 2024 October Q1
8 marks Moderate -0.8
    1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
    2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
  2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]
SPS SPS FM 2024 October Q6
6 marks Standard +0.8
Given that the equation $$2\log_2 x = \log_2(kx - 1) + 3,$$ only has one solution, find the value of \(x\). [6]
SPS SPS SM 2023 October Q6
8 marks Standard +0.3
In part (ii) of this question you must show detailed reasoning.
  1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
  2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]
SPS SPS FM 2023 October Q9
12 marks Standard +0.3
  1. \includegraphics{figure_9} The shape ABC shown in the diagram is a student's design for the sail of a small boat. The curve AC has equation \(y = 2 \log_2 x\) and the curve BC has equation \(y = \log_2\left(x - \frac{3}{2}\right) + 3\). State the x-coordinate of point A. [1]
  2. Determine the x-coordinate of point B. [3]
  3. By solving an equation involving logarithms, show that the x-coordinate of point C is 2. [4] It is given that, correct to 3 significant figures, the area of the sail is 0.656 units\(^2\).
  4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines. [4]
SPS SPS FM 2024 October Q5
9 marks Standard +0.3
In this question you must show detailed reasoning. The polynomial \(f(x)\) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of \(f(x)\). [1]
    2. Hence find the exact roots of the equation \(f(x) = 0\). [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form \(f(x) = 0\). [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS SM 2024 October Q9
9 marks Moderate -0.3
In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of f(x). [1]
    2. Hence find the exact roots of the equation f(x) = 0. [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS SM 2024 October Q7
6 marks Moderate -0.3
A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below. \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\) \(4\log_3 x - 3 \log_3 x - 2 = 0\) \(\log_3 x - 2 = 0\) \(\log_3 x = 2\) \(x = 8\)
  1. Identify the two mistakes that the student has made. [2]
  2. Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]
SPS SPS SM 2025 October Q7
7 marks Standard +0.3
In this question you must show detailed reasoning. Solve the following equations.
  1. \(y^6 + 7y^3 - 8 = 0\) [3]
  2. \(9^{x+1} + 3^x = 8\) [4]
SPS SPS SM 2025 October Q11
9 marks Moderate -0.8
A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.
  1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]
The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.
\(t\) minutes813213550
\(M\) kg0.40.30.20.10.05
The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.
    1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
    2. Interpret the value of \(k\) in context. [1]
  2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
  3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
  4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]
SPS SPS FM 2026 November Q1
6 marks Easy -1.2
  1. Solve the equation $$x\sqrt{2} - \sqrt{18} = x$$ writing the answer as a surd in simplest form. [3]
  2. Solve the equation $$4^{3x-2} = \frac{1}{2\sqrt{2}}$$ [3]
SPS SPS FM 2026 November Q4
6 marks Standard +0.3
  1. The curves \(e^x - 2e^y = 1\) and \(2e^x + 3e^{2y} = 41\) intersect at the point \(P\). Show that the \(y\)-coordinate of \(P\) satisfies the equation \(3e^{2y} + 4e^y - 39 = 0\). [1]
  2. In this question you must show detailed reasoning. Hence find the exact coordinates of \(P\). [5]