Questions: Logarithms
Before attempting these questions, it is highly recommended that you read Guide: Introduction to Logarithms.
Q1
For the following, find the value of \(x\), representing your answer exactly (not decimals).
1.1. \(\quad \log_{7}(x) = 1\)
1.2. \(\quad \log_{8}(x) = 3\)
1.3. \(\quad \log_{12}(x) = 0\)
1.4. \(\quad \log_{10}(100) = x\)
1.5. \(\quad \log_{2}(64) = x\)
1.6. \(\quad \log_{4}(2) = x\)
1.7. \(\quad \log_{3}(27) = x\)
1.8. \(\quad \log_{10}(1) = x\)
1.9. \(\quad \log_{x}(16) = 4\)
1.10. \(\quad \log_{x}(49) = 2\)
1.11. \(\quad \log_{x}(13) = 4\)
1.12. \(\quad \log_{2x}(12) = -1\)
Q2
Before attempting this question, write out the five laws of logarithms next to their names: the product rule, the quotient rule, the power rule, the zero rule, the identity rule.
Using the five laws of logarithms, find the value of \(x\):
2.1. \(\quad \log_{3}\left(\dfrac{1}{27}\right) = x\)
2.2. \(\quad 4\log_{4}(2) = x\)
2.3. \(\quad \log_{5}(10) + \log_{5}\left(\dfrac{5}{2}\right)=x\)
2.4. \(\quad 3\log_{7}\left(a^{1/3}\right) - \frac{1}{2}\log_{7}(a^2) = x\)
2.5. \(\quad \log_{x}(YZ) = M\)
2.6. \(\quad \log_{a}\left(y\right) - \log_a(x) = 11\)
Q3
Using the change of base rule and other laws of logs if required, express the following logarithms as expressions involving a logarithm to the specified base. Give your answer as simply as possible, evaluating if you can.
3.1. \(\quad \log_3(25)\) to base \(5\)
3.2. \(\quad \log_{8}(3)\) to base \(16\)
3.3. \(\quad \log_{e}(10)\) to base \(1000\)
3.4. \(\quad \ln(27)\) to base \(3\)
3.5. \(\quad \log_{4}(8x)\) to base \(2\)
After attempting the questions above, please click this link to find the answers.
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.