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S4 Mathematics: Indices and Logarithms (Paper 2)
S4 Mathematics: Indices and Logarithms (Paper 2)
Ten original questions concentrate on logarithms: reading log base 2 of 8 as three, log base 5 of 1 as zero, log 1000 as three and log base 3 of 81 as four, then applying the log laws to combine sums and differences into five, one and three. Change of base gives three halves for log base 9 of 27, a base is recovered from log base x of 16 equals four, and the paper ends with the error of distributing a logarithm over a sum.
Direct answer Ask what power of the base gives the number, turn sums and differences of logs into products and quotients, and change base by dividing logs.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 25 |
| Difficulty mix | Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1 |
| Chapter | Chapter: S4 Mathematics: Indices and Logarithms |
| Level | Four basic, three advanced, two challenge and one error-repair item |
02 Questions (10 items)
- FoundationFind the value of log base 2 of 8.
- FoundationFind the value of log base 5 of 1.
- FoundationFind the value of log 1000, with base 10.
- FoundationFind the value of log base 3 of 81.
- IntermediateFind the value of log base 2 of 4 plus log base 2 of 8.
- IntermediateFind the value of log 2 plus log 5, with base 10.
- IntermediateFind the value of log base 2 of 40 minus log base 2 of 5.
- ChallengeFind the value of log base 9 of 27 as a fraction.
- ChallengeGiven log base x of 16 equals 4, find x.
- Error redoError repair: a student writes log of (4 plus 6) as log 4 plus log 6. What is the correct value of log 10?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 3 | Two cubed is eight, so the logarithm is three. |
| 2 | 0 | The logarithm of one is zero for any base, since five to the zero is one. |
| 3 | 3 | Ten cubed is one thousand. |
| 4 | 4 | Three to the fourth is eighty-one. |
| 5 | 5 | Adding logs of the same base is the log of the product, and log base 2 of 32 is five. |
| 6 | 1 | The product is ten, whose base-ten logarithm is one. |
| 7 | 3 | Subtracting logs of the same base is the log of the quotient, and 40 over 5 is 8. |
| 8 | 3/2 | Nine to the three halves is twenty-seven, or change the base and divide the logs. |
| 9 | x equals 2 | Rewrite as x to the fourth equals sixteen, and x is positive, so it is two. |
| 10 | 1 | Logarithms do not distribute over addition; log 10 is one while the sum of the logs is about 1.38. |
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.
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.
