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S4 Maths: Logarithms Paper 4 (Error Review)
S4 Maths: Logarithms Paper 4 (Error Review)
This paper reworks the slips from papers 1 to 3: same-base addition, change of base, index equations, pH and decibels.
Quick answer 1. 7 | 2. 6 (not log2 20) | 3. 3 | 4. 2 | 5. x = 32 | 6. 3.322 | 7. 5.32 | 8. x = 4 | 9. pH 4, 10^-3 mol/L | 10. 30 dB
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 questions |
| Suggested time | 25 minutes |
| Difficulty mix | Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1 |
| Chapter | Chapter: S4 Maths — Logarithms |
| Level | Error review |
02 Questions (10 items)
- FoundationEvaluate log2 8 + log3 81.
- FoundationEvaluate log2 16 + log2 4 and explain why it is not log2 20.
- FoundationEvaluate log2 32 – log2 4.
- FoundationSimplify 2 log3 6 – log3 4.
- IntermediateSolve log2 x = 5.
- IntermediateUse change of base to find log2 10 to three decimal places.
- IntermediateSolve 2^x = 40 to two decimal places.
- ChallengeSolve log2 x + log2 (x – 2) = 3.
- ChallengeA solution has [H+] = 1 x 10^-4 mol/L; find its pH, and find [H+] when pH = 3.
- Error redoA student sees intensity 1000 times greater and says loudness rises by 1000 dB. What is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 3 + 4 = 7. | Simplify each log to an integer first, then add. |
| 2 | 4 + 2 = 6; adding logs multiplies the numbers, so log2(16×4) = log2 64 = 6. | Same-base addition multiplies the numbers (log2 20 is about 4.32). |
| 3 | 5 – 2 = 3. | Subtraction divides the numbers: log2(32/4) = log2 8 = 3. |
| 4 | log3(36/4) = log3 9 = 2. | Turn the coefficient into a power, then combine by division. |
| 5 | x = 2^5 = 32. | Rewrite as an index form; the number must stay positive. |
| 6 | log10 10 / log10 2 is about 3.322. | Change to base 10; the answer is a little above 3. |
| 7 | x = log10 40 / log10 2 is about 5.32. | Take logs of both sides to bring the power down. |
| 8 | x(x – 2) = 8 gives x = 4; x = -2 is rejected since logs need positive numbers. | Combine into one log, solve the quadratic, then check the domain. |
| 9 | pH = 4; [H+] = 1 x 10^-3 mol/L. | Each drop of 1 in pH means ten times the hydrogen ion concentration. |
| 10 | 10 log10 1000 = 30, so the rise is 30 dB. | Decibels are logarithmic, so a thousandfold intensity is only 30 dB. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
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