Inverse function with exponentials

Find inverse functions or rearrange exponential equations to express one variable in terms of another, such as expressing y in terms of x given x = f(y).

8 questions · Moderate -0.9

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CAIE P3 2006 June Q1
3 marks Easy -1.2
1 Given that \(x = 4 \left( 3 ^ { - y } \right)\), express \(y\) in terms of \(x\).
OCR C3 2006 June Q4
6 marks Moderate -0.3
4 It is given that \(y = 5 ^ { x - 1 }\).
  1. Show that \(x = 1 + \frac { \ln y } { \ln 5 }\).
  2. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  3. Hence find the exact value of the gradient of the curve \(y = 5 ^ { x - 1 }\) at the point (3, 25).
OCR MEI C3 Q5
4 marks Moderate -0.8
5 Given that \(x\) and \(t\) are related by the formula \(x = x _ { 0 } \mathrm { e } ^ { - 3 t }\), show that \(t = \ln \left( \frac { a } { x } \right) ^ { b }\) where \(a\) and \(b\) are to be determined.
OCR MEI C2 2012 January Q7
3 marks Moderate -0.8
7 Given that \(y = a + x ^ { b }\), find \(\log _ { 10 } x\) in terms of \(y\), \(a\) and \(b\).
OCR MEI C2 2011 June Q8
3 marks Easy -1.2
8 Using logarithms, rearrange \(p = s t ^ { n }\) to make \(n\) the subject.
Edexcel Paper 2 2019 June Q1
3 marks Moderate -0.8
  1. Given
$$2 ^ { x } \times 4 ^ { y } = \frac { 1 } { 2 \sqrt { 2 } }$$ express \(y\) as a function of \(x\).
OCR MEI AS Paper 2 2022 June Q3
3 marks Easy -1.2
3 You are given that \(y = A e ^ { 0.02 t }\).
  • Make \(t\) the subject of the formula.
  • Find the value of \(t\) when \(y = 10 ^ { 8 }\) and \(A = 6.62 \times 10 ^ { 7 }\).
OCR MEI Paper 1 2024 June Q5
5 marks Moderate -0.8
5
  1. Make \(y\) the subject of the formula \(\log _ { 10 } ( y - k ) = x \log _ { 10 } 2\), where \(k\) is a positive constant.
  2. Sketch the graph of \(y\) against \(x\).