Answers: Solving equations involving logarithms
These are the answers to Questions: Solving equations involving logarithms
Please attempt the questions before reading these answers!
Throughout this answer sheet, the natural logarithm \(\log_e(x)\) is written as \(\ln(x)\).
Q1
1.1. Here, \(y=x^{1/4}\) and \(x = y^4\).
1.2. Here, \(y=x^3\) and \(x = y^{1/3}\).
1.3. Here, \(x = \sqrt{y^{1/2}x^3}\), \(y = x^4/z^6\), \(z = \sqrt[3]{x^2y^{-1/2}}\).
Q2
2.1. The solution to \(6\log_3(x)+\log_3(5)=9\) is \(x=\sqrt[6]{\frac{3^9}{5}}\).
2.2. The solution to \(\log_2(16x)=6\) is \(x=4\).
2.3. The solution to \(\log_{12}e^{2t} = 4\) is \(t = 2\ln(12) = \ln(144)\).
2.4. The solution to \(log_9(x)+log_3(3x)=6\) is \(x=3^{10/3}\).
2.5. The solution to \(4\ln\sqrt{x} - \ln(1-2x) = 0\) is \(x = -1 + \sqrt{2}\).
2.6. The solution to \(\ln(x+1) - \ln(x) = e\) is \(x = \dfrac{1}{e^e - 1}\).
2.7. There are no solutions to \(\log_{10}(2y + 10) = \log_{10}(y-2)\).
2.8. The solutions to \(\log_3\sqrt{x} - \log_9 \sqrt{4x - 3} = 0\) are \(x = 1\) and \(x = 3\).
2.9. The solutions to \(\log_3(2-3x) = \log_9(6x^2 - 19x + 2)\) are \(x = -1/3\) and \(x= -2\).
2.10. The solutions to \(\log_3(x) - 2\log_x(3) = 1\) are \(x = 9\) and \(x = 1/3\).
Q3
The solutions are \(x = 15\) and \(y = 1/2\).
Version history and licensing
v1.0: initial version created 08/23 by Ellie Gurini as part of a University of St Andrews STEP project, and updated 10/25 by tdhc.