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S5 Maths paper 4: error drills on indices and logarithms
S5 Maths paper 4: error drills on indices and logarithms
Ten items target the four commonest slips: adding indices instead of multiplying, reversing fractional indices, splitting a logarithm of a sum, and ignoring the positive domain.
Direct answers Items 01 to 03 recompute index laws and fractional indices, 04 to 06 handle logarithm values, sums and same-base equations, 07 to 09 solve a combined logarithm, population growth and half-life, and 10 corrects splitting log(a + b).
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 25 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S5 Maths Exponential and Logarithmic Functions |
| Level | S5 |
02 Questions (10 items)
- FoundationSimplify 2 cubed times 2 to the fourth power and give its value.
- FoundationFind the value of 27 to the power of two thirds.
- FoundationSolve 3 to the power x equals 81.
- IntermediateFind log base 2 of 32 and log base 3 of one ninth.
- IntermediateFind log 8 plus log 125 (base 10).
- IntermediateSolve 2 to the power of (x + 1) equals 32.
- ChallengeSolve log base 2 of (x minus 1) plus log base 2 of (x plus 1) equals 3.
- ChallengeA city has 500 000 people growing 4% a year. Find the least number of years until it passes 700 000.
- ChallengeA substance has a half-life of 20 years and starts at 800 g. Find the mass after 60 years and the time to fall to 25 g.
- Error redoA student writes log(4 + 5) = log 4 + log 5 and says both sides are 1.301. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 2 to the seventh power, which is 128. | For the same base, multiply by adding the indices instead of expanding. |
| 2 | 9 (take the cube root to get 3, then square). | The denominator of a fractional index is a root and the numerator is a power. |
| 3 | x = 4. | Write the right side as 3 to the fourth power, then compare indices. |
| 4 | 5 and minus 2. | One ninth is 3 to the power minus 2, so a logarithm may be negative. |
| 5 | log 1000, which is 3. | Adding logarithms of the same base multiplies the numbers: 8 x 125 = 1000. |
| 6 | x = 4. | With the same base the indices are equal, so x + 1 = 5. |
| 7 | x = 3. | Combine to (x – 1)(x + 1) = 8, take the positive root and check the domain x > 1. |
| 8 | 9 years. Year 8 gives 684 285 and year 9 gives 711 656. | Solve 500000 x 1.04^n > 700000; n > 8.6 so take 9. |
| 9 | 100 g after 60 years; 100 years to reach 25 g. | Each 20 years halves the mass: three halvings in 60 years, five halvings for 25 g. |
| 10 | Only products split: log 9 is about 0.954 while log 4 + log 5 = log 20 is about 1.301. | log(ab) equals log a + log b; a sum inside the bracket cannot be split. |
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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