LOGARITHM REVISION QUESTIONS WITH ANSWERS

High School Logarithm Questions

1. Evaluate \( \log_{10} 1000 \)
2. Simplify: \( \log 2 + \log 5 \)
3. If \( \log x = 2 \), find the value of \( x \)
4. Solve: \( \log_{2} x = 5 \)
5. Using tables, find \( \log_{10} 32.5 \)
6. Find \( \log_{10} 0.00345 \) using tables
7. Prove: \( \log(ab) = \log a + \log b \)
8. Simplify: \( \log 50 - \log 2 \)
9. Evaluate \( \log_{10}(100 \times 0.01) \)
10. Express \( \log 8 \) in terms of \( \log 2 \)
11. If \( \log x = a \), express \( x \) in terms of \( a \)
12. Simplify: \( 2\log 3 - \log 9 \)
13. Evaluate: \( \log_{10} \left(\frac{25}{4}\right) \)
14. Solve: \( \log(x^2) = 6 \)
15. If \( \log 2 = 0.3010 \), find \( \log 16 \)
16. Using log tables, find the value of \( 73.4 \times 0.052 \)
17. Find the number of digits in \( 2^{10} \)
18. Solve for \( x \): \( \log_{3}(x - 2) = 2 \)
19. Simplify: \( \log_5(125) \)
20. Evaluate: \( \log(10^{-3}) \)
21. If \( \log a = x \) and \( \log b = y \), express \( \log \left(\frac{a^2}{\sqrt{b}}\right) \) in terms of \( x \) and \( y \)
22. Find the antilog of 2.643
23. Using tables, calculate \( \frac{465 \times 0.0732}{9.52} \)
24. Evaluate: \( \log_{4} 64 \)
25. Solve for \( x \): \( \log_{x} 16 = 4 \)
26. Simplify: \( \log(1000) - \log(10) \)
27. If \( \log 7 = 0.8451 \), find \( \log 49 \)
28. Prove: \( \log_b a = \frac{\log a}{\log b} \)
29. Solve: \( \log_{2}(x+3) = \log_{2}(x-1) + 1 \)
30. Find \( x \) if \( \log_{10} x = -2.3 \)

Answers

1. 3
2. 1
3. 100
4. 32
5. 1.5119
6. −2.461
7. By law: \( \log(ab) = \log a + \log b \)
8. \( \log 25 = 1.6989 \)
9. 0
10. \( \log 8 = 3\log 2 \)
11. \( x = 10^a \)
12. 0
13. \( \log 25 - \log 4 = \log \frac{25}{4} \approx 0.3979 \)
14. \( x = 10^3 = 1000 \)
15. 1.2040
16. \( \log(73.4) + \log(0.052) = 1.865 + (-1.284) = 0.581 \Rightarrow \text{antilog } 0.581 \approx 3.81 \)
17. 4 digits (since \( \log(2^{10}) = \log(1024) \approx 3.01 \))
18. 11
19. 3
20. −3
21. \( 2x - \frac{1}{2}y \)
22. \( \text{antilog}(2.643) \approx 441.7 \)
23. \( \log(465) + \log(0.0732) - \log(9.52) = 2.668 + (-1.135) - 0.979 \approx 0.554 \Rightarrow \text{antilog} \approx 3.58 \)
24. 3
25. 2
26. 2
27. \( 2\log 7 = 2 \times 0.8451 = 1.6902 \)
28. \( \log_b a = \frac{\log a}{\log b} \) — Change of base law
29. \( x = 5 \)
30. \( x = 10^{-2.3} \approx 0.00501 \)