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

483 questions

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Edexcel C2 Q5
9 marks Moderate -0.3
  1. Describe fully a single transformation that maps the graph of \(y = 3^x\) onto the graph of \(y = (\frac{1}{3})^x\). [1]
  2. Sketch on the same diagram the curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\), showing the coordinates of any points where each curve crosses the coordinate axes. [3]
The curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) intersect at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt{2}\). [5]
Edexcel C2 Q5
7 marks Moderate -0.3
  1. Find the value of \(a\) such that $$\log_a 27 = 3 + \log_a 8.$$ [3]
  2. Solve the equation $$2^{x+3} = 6^{x-1},$$ giving your answer to 3 significant figures. [4]
OCR C2 Q5
8 marks Standard +0.3
  1. Solve the equation $$\log_2 (6 - x) = 3 - \log_2 x.$$ [4]
  2. Find the smallest integer \(n\) such that $$3^{n-2} > 8^{250}.$$ [4]
OCR C2 Q3
6 marks Moderate -0.8
During one day, a biological culture is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times (1.06)^t.$$ Using this model,
  1. find the number of bacteria present at 11 a.m., [2]
  2. find, to the nearest minute, the time when the initial number of bacteria will have doubled. [4]
OCR C3 Q3
6 marks Moderate -0.3
The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180e^{-0.017t}.$$
  1. Find the value of \(t\) for which the mass is 25 grams. [3]
  2. Find the rate at which the mass is decreasing when \(t = 55\). [3]
OCR C3 Q8
16 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]
OCR MEI C3 Q4
Moderate -0.8
The temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).
OCR MEI C3 2011 January Q5
8 marks Standard +0.3
  1. On a single set of axes, sketch the curves \(y = e^x - 1\) and \(y = 2e^{-x}\). [3]
  2. Find the exact coordinates of the point of intersection of these curves. [5]
Edexcel C3 Q4
10 marks Standard +0.3
  1. Find, as natural logarithms, the solutions of the equation $$e^{2x} - 8e^x + 15 = 0.$$ [4]
  2. Use proof by contradiction to prove that \(\log_5 3\) is irrational. [6]
Edexcel C3 Q2
7 marks Moderate -0.8
Solve each equation, giving your answers in exact form.
  1. \(e^{4x-3} = 2\) [3]
  2. \(\ln (2y - 1) = 1 + \ln (3 - y)\) [4]
OCR C3 Q4
6 marks Moderate -0.8
Solve each equation, giving your answers in exact form.
  1. \(\mathrm{e}^{4x-3} = 2\) [2]
  2. \(\ln(2y - 1) = 1 + \ln(3 - y)\) [4]
AQA C4 2016 June Q4
7 marks Moderate -0.3
The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where \(m_0\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.
    1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004, correct to six decimal places. [1 mark]
    2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
  2. The half-life of a radioactive substance is the time it takes for a mass of \(m_0\) to reduce to a mass of \(\frac{1}{2}m_0\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]
Edexcel AEA 2008 June Q5
14 marks Challenging +1.8
  1. Anna, who is confused about the rules for logarithms, states that $$(\log_3 p)^2 = \log_3 (p^2)$$ and $$\log_3(p + q) = \log_3 p + \log_3 q.$$ However, there is a value for \(p\) and a value for \(q\) for which both statements are correct. Find the value of \(p\) and the value of \(q\). [7]
  2. Solve $$\frac{\log_3(3x^3 - 23x^2 + 40x)}{\log_3 9} = 0.5 + \log_3(3x - 8).$$ [7]
OCR H240/03 2023 June Q1
3 marks Easy -1.2
Using logarithms, solve the equation $$4^{2x+1} = 5^x,$$ giving your answer correct to 3 significant figures. [3]
AQA AS Paper 1 2020 June Q6
9 marks Moderate -0.3
  1. It is given that $$f(x) = x^3 - x^2 + x - 6$$ Use the factor theorem to show that \((x - 2)\) is a factor of \(f(x)\). [2 marks]
  2. Find the quadratic factor of \(f(x)\). [1 mark]
  3. Hence, show that there is only one real solution to \(f(x) = 0\) [3 marks]
  4. Find the exact value of \(x\) that solves $$e^{3x} - e^{2x} + e^x - 6 = 0$$ [3 marks]
AQA AS Paper 2 2018 June Q12
8 marks Standard +0.3
Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.
  1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
  2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
  3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
  4. Comment on the long-term accuracy of the model. [1 mark]
AQA AS Paper 2 2020 June Q8
6 marks Moderate -0.3
  1. Using \(y = 2^{2x}\) as a substitution, show that $$16^x - 2^{(2x+3)} - 9 = 0$$ can be written as $$y^2 - 8y - 9 = 0$$ [2 marks]
  2. Hence, show that the equation $$16^x - 2^{(2x+3)} - 9 = 0$$ has \(x = \log_2 3\) as its only solution. Fully justify your answer. [4 marks]
AQA AS Paper 2 2023 June Q4
5 marks Moderate -0.3
Find the exact solution of the equation \(\ln(x + 1) + \ln(x - 1) = \ln 15 - 2\ln 7\) Fully justify your answer. [5 marks]
AQA Paper 1 2024 June Q14
10 marks Standard +0.3
  1. The equation $$x^3 = e^{6-2x}$$ has a single solution, \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 4 [2 marks]
  2. Show that the equation \(x^3 = e^{6-2x}\) can be rearranged to give $$x = 3 - \frac{3}{2}\ln x$$ [3 marks]
    1. Use the iterative formula $$x_{n+1} = 3 - \frac{3}{2}\ln x_n$$ with \(x_1 = 4\), to find \(x_2\), \(x_3\) and \(x_4\) Give your answers to three decimal places. [2 marks]
    2. Figure 1 below shows a sketch of parts of the graphs of $$y = 3 - \frac{3}{2}\ln x \quad \text{and} \quad y = x$$ On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place. Label, on the \(x\)-axis, the positions of \(x_2\), \(x_3\) and \(x_4\) [2 marks]
      [diagram]
    3. Explain why the iterative formula $$x_{n+1} = 3 - \frac{3}{2}\ln x_n$$ fails to converge to \(\alpha\) when the starting value is \(x_1 = 0\) [1 mark]
AQA Paper 1 Specimen Q10
10 marks Standard +0.3
The function f is defined by $$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$
  1. Using set notation, state the range of f [2 marks]
  2. The inverse of f is \(f^{-1}\)
    1. Using set notation, state the domain of \(f^{-1}\) [1 mark]
    2. Find an expression for \(f^{-1}(x)\) [3 marks]
  3. The function g is defined by $$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
    1. Find an expression for \(gf(x)\) [1 mark]
    2. Solve the equation \(gf(x) = 2\), giving your answer in an exact form. [3 marks]
AQA Paper 2 2024 June Q4
3 marks Moderate -0.8
Use logarithms to solve the equation $$5^{x-2} = 7^{1570}$$ Give your answer to two decimal places. [3 marks]
AQA Paper 3 2020 June Q5
9 marks Moderate -0.3
The number of radioactive atoms, \(N\), in a sample of a sodium isotope after time \(t\) hours can be modelled by $$N = N_0 e^{-kt}$$ where \(N_0\) is the initial number of radioactive atoms in the sample and \(k\) is a positive constant. The model remains valid for large numbers of atoms.
  1. It takes 15.9 hours for half of the sodium atoms to decay. Determine the number of days required for at least 90\% of the number of atoms in the original sample to decay. [5 marks]
  2. Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures. [2 marks]
  3. Explain why the model can only provide an estimate for the number of remaining atoms. [1 mark]
  4. Explain why the model is invalid in the long run. [1 mark]
AQA Paper 3 2024 June Q8
8 marks Moderate -0.3
The temperature \(\theta\) °C of an oven \(t\) minutes after it is switched on can be modelled by the equation $$\theta = 20(11 - 10e^{-kt})$$ where \(k\) is a positive constant. Initially the oven is at room temperature. The maximum temperature of the oven is \(T\) °C The temperature predicted by the model is shown in the graph below. \includegraphics{figure_8} \begin{enumerate}[label=(\alph*)] \item Find the room temperature. [2 marks] \item Find the value of \(T\) [2 marks] \item The oven reaches a temperature of 86 °C one minute after it is switched on.
  1. Find the value of \(k\). [2 marks]
  2. Find the time it takes for the temperature of the oven to be within 1°C of its maximum. [2 marks]
Edexcel AS Paper 1 Specimen Q12
4 marks Moderate -0.3
A student was asked to give the exact solution to the equation $$2^{2x+4} - 9(2^x) = 0$$ The student's attempt is shown below: $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ Let \(2^x = y\) $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$
  1. Identify the two errors made by the student. [2]
  2. Find the exact solution to the equation. [2]
OCR MEI AS Paper 2 2018 June Q12
10 marks Moderate -0.8
In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).
  1. Write down the values of \(A\) and \(k\). [2]
  2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
  3. What does the model suggest about the number of fruit flies in the long run? [1]
Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$
  1. Identify three ways in which this second model is consistent with the known data. [3]
    1. Identify one feature which is not accounted for by the second model. [1]
    2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]