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

483 questions

Sort by: Default | Easiest first | Hardest first
Edexcel C2 Q4
6 marks Standard +0.3
Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]
Edexcel C2 Q9
9 marks Standard +0.3
A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by $$P = \frac{2000a^t}{4 + a^t},$$ where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,
  1. calculate, to 4 decimal places, the value of \(a\), [4]
  2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800. [4]
  3. With reference to this model, give a reason why the population of deer cannot exceed 2000. [1]
Edexcel C2 Q28
10 marks Moderate -0.3
  1. Given that \(3 + 2 \log_2 x = \log_2 y\), show that \(y = 8x^2\). [3]
  2. Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log_2 x = \log_2 (14x - 3).$$ [3]
  3. Show that \(\log_2 \alpha = -2\). [1]
  4. Calculate \(\log_2 \beta\), giving your answer to 3 significant figures. [3]
Edexcel C2 Q34
6 marks Moderate -0.8
  1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
  2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 decimals places. [4]
Edexcel C3 Q18
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).
  1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]
The \(x\)-coordinate of the point of intersection of the graphs is \(\alpha\).
  1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
  2. Show that \(-1 < \alpha < 0\). [2]
The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).
  1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
  2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]
Edexcel C3 Q25
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2 (x^2 + 4x + 3) - \log_2 (x^2 + x) = 4\). [4]
OCR C1 2013 January Q2
6 marks Easy -1.3
Solve the equations
  1. \(3^n = 1\), [1]
  2. \(t^{-3} = 64\), [2]
  3. \((8p^6)^{\frac{1}{3}} = 8\). [3]
Edexcel C1 Q1
3 marks Moderate -0.8
Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]
OCR C1 Q1
4 marks Moderate -0.5
Find the value of \(y\) such that $$4^{y+1} = 8^{2y-1}.$$ [4]
OCR C1 Q6
10 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Given that \(y = 2^x\), find expressions in terms of \(y\) for
  1. \(2^{x+2}\), [2]
  2. \(2^{3-x}\). [2]
\item Show that using the substitution \(y = 2^x\), the equation $$2^{x+2} + 2^{3-x} = 33$$ can be rewritten as $$4y^2 - 33y + 8 = 0.$$ [2] \item Hence solve the equation $$2^{x+2} + 2^{3-x} = 33.$$ [4]
AQA C2 2009 June Q9
10 marks Moderate -0.8
    1. Find the value of \(p\) for which \(\sqrt{125} = 5^p\). [2]
    2. Hence solve the equation \(5^{2x} = \sqrt{125}\). [1]
  2. Use logarithms to solve the equation \(3^{2x-1} = 0.05\), giving your value of \(x\) to four decimal places. [3]
  3. It is given that $$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms. [4]
Edexcel C2 Q3
9 marks Standard +0.3
A population of deer is introduced into a park. The population P at t years after the deer have been introduced is modelled by $$P = \frac{2000a^t}{4 + a^t},$$ where a is a constant. Given that there are 800 deer in the park after 6 years,
  1. calculate, to 4 decimal places, the value of a, [4]
  2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800. [4]
  3. With reference to this model, give a reason why the population of deer cannot exceed 2000. [1]
Edexcel C2 Q4
7 marks Moderate -0.8
Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after \(x\) years, the total value of an initial £1 investment is £\(y\), where $$y = 1.04^x.$$
  1. Sketch the graph of \(y = 1.04^x\), \(x \geq 0\). [2]
  2. Calculate, to the nearest £, the total value of an initial £800 investment after 10 years. [2]
  3. Use logarithms to find the number of years it takes to double the total value of any initial investment. [3]
Edexcel C2 Q3
6 marks Moderate -0.3
  1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
  2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 d. p. [4]
OCR C2 2007 January Q9
10 marks Standard +0.3
On its first trip between Maltby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses 2% more coal than the previous trip.
  1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures. [2]
  2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02^N < 1.52.$$ [4]
  3. Hence, by using logarithms, find the greatest number of trips possible. [4]
OCR C2 Specimen Q8
10 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the curve \(y = 1.25^x\).
  1. A point on the curve has y-coordinate 2. Calculate its x-coordinate. [3]
  2. Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line \(x = 4\). [4]
  3. State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate. [2]
  4. Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region. [1]
OCR MEI C2 2010 January Q9
5 marks Moderate -0.8
  1. Sketch the graph of \(y = 3^x\). [2]
  2. Use logarithms to solve \(3^{2x+1} = 10\), giving your answer correct to 2 decimal places. [3]
OCR MEI C2 2013 January Q8
5 marks Moderate -0.8
  1. Sketch the graph of \(y = 3^x\). [2]
  2. Solve the equation \(3^{3x-1} = 500000\). [3]
OCR MEI C2 2006 June Q9
4 marks Moderate -0.8
Use logarithms to solve the equation \(5^{3x} = 100\). Give your answer correct to 3 decimal places. [4]
OCR MEI C2 2008 June Q9
3 marks Moderate -0.8
Use logarithms to solve the equation \(5^x = 235\), giving your answer correct to 2 decimal places. [3]
OCR MEI C2 2013 June Q11
11 marks Moderate -0.3
A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.
  1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
  2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
  3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
  4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]
OCR MEI C2 2014 June Q10
4 marks Moderate -0.3
Use logarithms to solve the equation \(3^{x+1} = 5^{2x}\). Give your answer correct to 3 decimal places. [4]
OCR MEI C2 2016 June Q8
5 marks Moderate -0.8
  1. Simplify \(\log_a 1 - \log_a (a^m)^3\). [2]
  2. Use logarithms to solve the equation \(3^{2x+1} = 1000\). Give your answer correct to 3 significant figures. [3]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Evaluate $$\log_3 27 - \log_3 4.$$ [4]
  2. Solve the equation $$4^x - 3(2^{x+1}) = 0.$$ [5]
Edexcel C2 Q5
8 marks Moderate -0.8
  1. Sketch the curve \(y = 5^{x-1}\), showing the coordinates of any points of intersection with the coordinate axes. [2]
  2. Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5^{x-1}\) intersects
    1. the straight line \(y = 10\),
    2. the curve \(y = 2^x\). [6]