Exponential relation to line equation

Given an exponential relation like a^y = b^(cx+d), prove by taking logarithms that the graph of y against x is a straight line and find its gradient or intercept.

7 questions · Moderate -0.5

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CAIE P2 2024 November Q1
5 marks Moderate -0.3
1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), where \(a\) and \(k\) are constants.
The graph of \(y\) against \(x\) is a straight line.
  1. Use logarithms to show that the gradient of the straight line is \(\frac { 3 } { 2 \ln a }\).
  2. Given that the straight line passes through the points \(( 0.4,0.95 )\) and \(( 3.3,3.80 )\), find the values of \(a\) and \(k\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-03_2723_33_99_21}
CAIE P2 2024 November Q1
5 marks Moderate -0.3
1 The variables \(x\) and \(y\) satisfy the equation \(a ^ { 2 y } = \mathrm { e } ^ { 3 x + k }\), where \(a\) and \(k\) are constants.
The graph of \(y\) against \(x\) is a straight line.
  1. Use logarithms to show that the gradient of the straight line is \(\frac { 3 } { 2 \ln a }\).
  2. Given that the straight line passes through the points \(( 0.4,0.95 )\) and \(( 3.3,3.80 )\), find the values of \(a\) and \(k\). \includegraphics[max width=\textwidth, alt={}, center]{468efb3f-be7b-4f9e-b8c3-c6fd40d7714a-03_2723_33_99_21}
CAIE P2 2007 June Q2
6 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { x + 2 }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. Find the exact value of the gradient of this line.
  2. Calculate the \(x\)-coordinate of the point of intersection of this line with the line \(y = 2 x\), giving your answer correct to 2 decimal places.
CAIE P2 2013 June Q4
4 marks Moderate -0.8
4 The variables \(x\) and \(y\) satisfy the equation \(5 ^ { y + 1 } = 2 ^ { 3 x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line.
  2. Find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the \(y\)-axis.
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { 2 - x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
  2. Calculate the exact \(x\)-coordinate of the point of intersection of this line with the line with equation \(y = 2 x\), simplifying your answer.
CAIE P3 2024 June Q3
5 marks Moderate -0.3
3 The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { 2 \mathrm { y } - 1 } = \mathrm { b } ^ { \mathrm { x } - \mathrm { y } }\), where \(a\) and \(b\) are constants.
  1. Show that the graph of \(y\) against \(x\) is a straight line.
  2. Given that \(\mathrm { a } = \mathrm { b } ^ { 3 }\), state the equation of the straight line in the form \(\mathrm { y } = \mathrm { px } + \mathrm { q }\), where \(p\) and \(q\) are rational numbers in their simplest form.
CAIE P3 2020 November Q3
5 marks Moderate -0.3
3 The variables \(x\) and \(y\) satisfy the relation \(2 ^ { y } = 3 ^ { 1 - 2 x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
  2. Find the exact \(x\)-coordinate of the point of intersection of this line with the line \(y = 3 x\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.