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

483 questions

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OCR H240/03 2018 December Q1
3 marks Moderate -0.8
Use logarithms to solve the equation \(2^{3x-1} = 3^{x+4}\), giving your answer correct to 3 significant figures. [3]
OCR H240/01 2017 Specimen Q5
4 marks Moderate -0.3
In this question you must show detailed reasoning. Use logarithms to solve the equation \(3^{2x+1} = 4^{100}\), giving your answer correct to 3 significant figures. [4]
Pre-U Pre-U 9794/1 2010 June Q1
3 marks Easy -1.2
Solve the equation \(2^x = 4^{2x+1}\). [3]
Pre-U Pre-U 9794/1 2011 June Q4
6 marks Moderate -0.8
  1. Show that \(4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)\), where \(x > \frac{2}{3}\). [3]
  2. Hence solve the equation \(4 \ln x - \ln(3x - 2) - \ln x^2 = 0\). [3]
Pre-U Pre-U 9794/2 2011 June Q5
7 marks Moderate -0.8
Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3e^{-0.02t}\) units and the concentration of Coldcure is \(5e^{-0.07t}\) units. Each drug becomes ineffective when its concentration falls below 1 unit.
  1. Show that Coldcure becomes ineffective before Antiflu. [3]
  2. Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
  3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]
Pre-U Pre-U 9794/2 2012 June Q4
4 marks Easy -1.2
Use logarithms to solve the equation \(2^{2x-1} = 5\). [4]
Pre-U Pre-U 9794/2 2016 June Q2
4 marks Easy -1.2
Solve the equation \(4 \times 3^x = 5\), giving the solution in an exact form. [4]
Edexcel AEA 2015 June Q1
6 marks Moderate -0.5
  1. Sketch the graph of the curve with equation $$y = \ln(2x + 5), \quad x > -\frac{5}{2}$$ On your sketch you should clearly state the equations of any asymptotes and mark the coordinates of points where the curve meets the coordinate axes. [3]
  2. Solve the equation \(\ln(2x + 5) = \ln 9\) [3]