Answers: Solving exponential equations

Summary

Answers to questions relating to solving exponential equations.

These are the answers to Questions: Solving exponential equations

Please attempt the questions before reading these answers!

Throughout this answer sheet, the natural logarithm \(\log_e(x)\) is written as \(\ln(x)\).

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1. The solution to \(\sqrt[4]{x-4} = 5\) is \(x = 629\).

2. The solution to \(x^4=2^8\) is \(x = 4\).

3. The solution to \(11^x=121^{x-1}\) is \(x = 2\).

4. The solution to \(x^{0.5}\) is \(x = 529\).

5. The solution to \(8^{2-x}=2^{4+3x}\) is \(x = \dfrac{1}{3}\).

6. The solution to \(2^{3x} = 10\) is \(x = \dfrac{\log_2(10)}{3}\).

7. The solution to \(5^{3-x}=625\) is \(x = -1\).

8. The solution to \(16^{2x}=4^{x-1}\) is \(x = -\dfrac{1}{3}\).

9. The solution to \(7^{2-x}=4^{2x+3}\) is \(x = \log_{112}\left(\dfrac{49}{64}\right)\).

10. The solution to \(16=8^{3-7x}\) is \(x = \dfrac{5}{21}\).

11. The solution to \(e^{3-8x}-9=0\) is \(x = \dfrac{3-\ln(9)}{8}\).

12. The solution to \(e^{4-3x}+8=12\) is \(x = \frac{4-\ln(4)}{3}\).

13. The solution to \(\sqrt[3]{2^{4x}-4}=5\) is \(x = \dfrac{\log_2(129)}{4}\).

14. The solution to \(\sqrt[3]{e^{2x}-13}=81^{\frac{1}{4}}\) is \(x = \dfrac{\ln(40)}{2}\).

15. The solution to \(\dfrac{5xa^{-7}b^{9}}{9a^2b^{-10}} = \dfrac{25b^{19}}{3a^9}\) is \(x = 15\).

16. The solution to \(4^x\cdot 2^x=64\) is \(x = 2\).

17. The solution to \({\dfrac{5^{x+1}\cdot6^{x+1}}{3^{x+1}}}=100\) is \(x = 1\).

18. The solution to \(\quad \dfrac{\left[\left(\frac{1}{2}\right)^x\cdot\left(\frac{-1}{4}\right)^x\right]}{\left(\frac{2}{3}\right)^x}=-\frac{27}{4096}\) is \(x = 3\).

19. The solution to \(3^{x+1}=7^x\) is \(x = \log_{7/3}(3)\).

20. The solution to \(5^{x+1}+5^x=12\) is \(x = \log_5(2)\).

21. The solution to \(2^{3x-1}=10^x\) is \(x = \log_{4/5}(2)\).

22. The solution to \(2^{2x}-2^{x+3}-2^4=0\) is \(x = \log_2(4+4\sqrt{2})\).

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Version history and licensing

v1.0: initial version created 08/23 by Zoë Gemmell, Isabella Lewis, Akshat Srivastava as part of a University of St Andrews STEP project.

• v1.1: edited 05/24 by tdhc.

This work is licensed under CC BY-NC-SA 4.0.