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S5 Maths: Indices and Logarithms: Paper 2 (intermediate)
S5 Maths: Indices and Logarithms: Paper 2 (intermediate)
Ten items applying the index laws to fractional and negative indices and simplification, plus logarithm addition and subtraction and simple exponential equations.
Direct answers Items 1-3: 9, 1/4, 3. Items 4-6: 5, 1, 3. Items 7-9: x = 5, x = 3, a cubed times b to the fifth. Item 10: wrong; the answer is 3.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S5 Maths: Indices and Logarithms |
| Level | Intermediate |
02 Questions (10 items)
- FoundationEvaluate 27 to the power two thirds.
- FoundationEvaluate 8 to the power negative two thirds.
- FoundationFind the logarithm of 125 to base 5.
- IntermediateEvaluate the logarithm of 8 to base 2 plus the logarithm of 4 to base 2.
- IntermediateWithout a calculator, evaluate the logarithm of 2 to base 10 plus the logarithm of 5 to base 10.
- IntermediateEvaluate the logarithm of 54 to base 3 minus the logarithm of 2 to base 3.
- ChallengeSolve 2 to the power x equals 32.
- ChallengeSolve 3 to the power of x plus 1 equals 81.
- ChallengeSimplify the square of a squared b cubed, all divided by a b, where a and b are not zero.
- Error redoA student adds 1 to the base 10 logarithm of 100 and writes it as the logarithm of 110. What is wrong, and what is the correct value?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 9 | Take the cube root first to get 3, then square it to get 9. |
| 2 | 1/4 | The two thirds power of 8 is 4, and the negative index makes it 1/4. |
| 3 | 3 | Since 5 cubed is 125, the logarithm to base 5 is 3. |
| 4 | 5 | Adding logarithms multiplies the numbers, and 8 times 4 is 32 whose base 2 logarithm is 5. |
| 5 | 1 | The numbers multiply to 10, and the base 10 logarithm of 10 is 1. |
| 6 | 3 | Subtracting logarithms divides the numbers, and 54 divided by 2 is 27 whose base 3 logarithm is 3. |
| 7 | x = 5 | Since 32 is 2 to the power 5, x equals 5. |
| 8 | x = 3 | Since 81 is 3 to the power 4, x plus 1 equals 4 and x equals 3. |
| 9 | a to the power 3 times b to the power 5 | Square the bracket first to get a to the 4 and b to the 6, then subtract the indices of each letter. |
| 10 | That is wrong; the answer is 3 | Logarithms cannot be combined that way: the two terms are 2 and 1, so the sum is 3. |
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.
Before adding or subtracting logarithms, rewrite the numbers as a product or a quotient.Tutor Registration →
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