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

483 questions

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CAIE P2 2015 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(| 3 x + 4 | = | 3 x - 11 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 2 ^ { y } + 4 \right| = \left| 3 \times 2 ^ { y } - 11 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2016 June Q3
6 marks Moderate -0.3
3 Given that \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\), find the possible values of \(\mathrm { e } ^ { x }\) and hence solve the equation \(3 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x } = 14\) correct to 3 significant figures.
CAIE P2 2016 June Q1
3 marks Moderate -0.8
1 Given that \(5 ^ { 3 x } = 7 ^ { 4 y }\), use logarithms to find the value of \(\frac { x } { y }\) correct to 4 significant figures.
CAIE P2 2017 June Q1
3 marks Moderate -0.8
1 Given that \(5 ^ { x } = 3 ^ { 4 y }\), use logarithms to show that \(y = m x\) and find the value of the constant \(m\) correct to 3 significant figures.
CAIE P2 2017 June Q2
4 marks Moderate -0.8
2 Use logarithms to solve the equation \(3 ^ { x + 4 } = 5 ^ { 2 x }\), giving your answer correct to 3 significant figures.
CAIE P2 2018 June Q1
5 marks Moderate -0.3
1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).
CAIE P2 2018 June Q4
7 marks Standard +0.3
4
  1. Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2\).
  2. Hence solve the equation $$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$ giving the value of \(u\) correct to 4 significant figures.
CAIE P3 2004 June Q4
6 marks Moderate -0.3
4
  1. Show that if \(y = 2 ^ { x }\), then the equation $$2 ^ { x } - 2 ^ { - x } = 1$$ can be written as a quadratic equation in \(y\).
  2. Hence solve the equation $$2 ^ { x } - 2 ^ { - x } = 1$$
CAIE P3 2007 June Q4
6 marks Standard +0.3
4 Using the substitution \(u = 3 ^ { x }\), or otherwise, solve, correct to 3 significant figures, the equation $$3 ^ { x } = 2 + 3 ^ { - x }$$
CAIE P3 2008 June Q2
5 marks Standard +0.3
2 Solve, correct to 3 significant figures, the equation $$\mathrm { e } ^ { x } + \mathrm { e } ^ { 2 x } = \mathrm { e } ^ { 3 x }$$
CAIE P3 2009 June Q1
4 marks Moderate -0.8
1 Solve the equation \(\ln \left( 2 + \mathrm { e } ^ { - x } \right) = 2\), giving your answer correct to 2 decimal places.
CAIE P3 2010 June Q1
4 marks Moderate -0.8
1 Solve the equation $$\frac { 2 ^ { x } + 1 } { 2 ^ { x } - 1 } = 5$$ giving your answer correct to 3 significant figures.
CAIE P3 2011 June Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$
  1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
  2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.
CAIE P3 2011 June Q2
5 marks Moderate -0.8
2
  1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
  2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$
CAIE P3 2011 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 2 x - 1 } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q1
3 marks Moderate -0.3
1 Solve the equation \(\left| 4 - 2 ^ { x } \right| = 10\), giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q2
4 marks Moderate -0.3
2 Solve the equation \(\ln ( 2 x + 3 ) = 2 \ln x + \ln 3\), giving your answer correct to 3 significant figures.
CAIE P3 2014 June Q1
3 marks Moderate -0.5
1 Solve the equation \(\log _ { 10 } ( x + 9 ) = 2 + \log _ { 10 } x\).
CAIE P3 2014 June Q4
7 marks Standard +0.3
4 The equation \(x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }\) has one positive real root, denoted by \(\alpha\).
  1. Show that \(\alpha\) lies between \(x = 1\) and \(x = 2\).
  2. Show that if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2015 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(2 ^ { 5 x } = 3 ^ { 2 x + 1 }\), giving the answer correct to 3 significant figures.
CAIE P3 2015 June Q2
4 marks Moderate -0.8
2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\ln ( x + 4 ) = 2 \ln x + \ln 4\), giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(2 | x - 1 | = 3 | x |\).
  2. Hence solve the equation \(2 \left| 5 ^ { x } - 1 \right| = 3 \left| 5 ^ { x } \right|\), giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q6
7 marks Standard +0.3
6
  1. By sketching a suitable pair of graphs, show that the equation $$5 \mathrm { e } ^ { - x } = \sqrt { } x$$ has one root.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.