Answers: Logarithms
These are answers to: Questions: 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. \(\quad \log_{7}(x) = 1\) rearranged gives \(7^1=x\) so \(x=7\).
1.2. \(\quad \log_{8}(x) = 3\) rearranged gives \(8^3=x\) so \(x=512\).
1.3. \(\quad \log_{12}(x) = 0\) rearranged gives \(12^0=x\) so \(x=1\).
1.4. \(\quad \log_{10}(100) = x\) rearranged gives \(10^x=100\) so \(x=2\).
1.5. \(\quad \log_{2}(64) = x\) rearranged gives \(2^x=64\) so \(x=6\).
1.6. \(\quad \log_{4}(2) = x\) rearranged gives \(4^x=2\) so \(x=\dfrac{1}{2}\).
1.7. \(\quad \log_{3}(27) = x\) rearranged gives \(3^x=27\) so \(x=3\).
1.8. \(\quad \log_{10}(1) = x\) rearranged gives \(10^x=1\) so \(x=0\).
1.9. \(\quad \log_{x}(16) = 4\) rearranged gives \(x^4=16\) so \(x=\sqrt[4]{16}=2\).
1.10. \(\quad \log_{x}(49) = 2\) rearranged gives \(x^2=49\) so \(x=\sqrt{49}=7\).
1.11. \(\quad \log_{x}(13) = 4\) rearranged gives \(x^4=13\) so \(x=\sqrt[4]{13}\).
1.12. \(\quad \log_{2x}(12) = -1\) rearranged gives \((2x)^{-1}=12\) so \(x=\dfrac{1}{24}\).
Q2
The product rule: \(\log_{a}(M\cdot N)=\log_{a}(M)+\log_{a}(N)\)
The quotient rule: \(\log_{a}\left(\dfrac{M}{N}\right)=\log_{a}(M)-\log_{a}(N)\)
The power rule: \(\log_{a}(M^k)=k\cdot\log_{a}(M)\)
The zero rule: \(\log_{a}(1)=0\)
The identity rule: \(\log_{a}(a)=1\)
2.1. The solution to \(\log_{3}(\dfrac{1}{27}) = x\) is \(x = -1/3\).
2.2. The solution to \(4\log_{4}(2) = x\) is \(x = 2\).
2.3. The solution to \(\log_{5}(10) + \log_{5}\left(\dfrac{5}{2}\right)=x\) is \(x = 2\).
2.4. The solution to \(3\log_{7}\left(a^{1/3}\right) - \frac{1}{2}\log_{7}(a^2) = x\) is \(x = 0\).
2.5. The solution to \(\log_{x}(YZ) = M\) is \(x=\sqrt[M]{YZ}\).
2.6. The solution to \(\log_{a}\left(y\right) - \log_a(x) = 11\) is \(x = ya^{-11}\).
Q3
3.1. \(\log_3(25)\) is equal to \(\dfrac{2}{\log_5(3)}\).
3.2. \(\log_{8}(3)\) is equal to \(\dfrac{4\log_{16}(3)}{3}\).
3.3. \(\log_{e}(10)\) is equal to \(\dfrac{1}{\log_{1000}(e^3)}\).
3.4. \(\ln(27)\) is equal to \(\dfrac{3}{\log_{3}(e)}\).
3.5. \(\log_{4}(8x)\) is equal to \(\dfrac{3}{2} + \log_2\left(\sqrt{2}\right)\).
Version history and licensing
v1.0: initial version created 08/23 by Zoë Gemmell as part of a University of St Andrews STEP project.
- v1.1: edited 05/24 by tdhc.