中文（繁體）
中文（简体）
English
S5 Maths: Indices and Logarithms: Paper 3 (challenge)
S5 Maths: Indices and Logarithms: Paper 3 (challenge)
Ten items reaching logarithm equations and applications: rewriting in exponential form, combining logarithms, quadratic substitution, matching bases and doubling contexts.
Direct answers Items 1-3: x = 13, x = 13, x = 0 or 2. Items 4-6: x = 5, 5, 1.3010. Items 7-9: x = 3, x = 6, 3 hours. Item 10: negative 3 is rejected, so only x = 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 | Challenge |
02 Questions (10 items)
- FoundationSolve the logarithm of x plus 3 to base 2 equals 4.
- FoundationSolve the logarithm of 2x minus 1 to base 5 equals 2.
- FoundationSolve 2 to the power 2x minus 5 times 2 to the power x plus 4 equals 0.
- IntermediateSolve the base 10 logarithm of x plus the base 10 logarithm of x minus 3 equals 1.
- IntermediateEvaluate the base 2 logarithm of 12 minus the base 2 logarithm of 3, then add 3.
- IntermediateGiven the base a logarithm of 2 is 0.3010 and of 5 is 0.6990, find the base a logarithm of 20.
- ChallengeSolve 9 to the power x equals 27 to the power x minus 1.
- ChallengeSolve 8 to the power x equals 4 to the power x plus 3.
- ChallengeA colony of bacteria doubles every hour. It starts with 500 and grows to 4000. How many hours does it take?
- Error redoA student solves the base 2 logarithm of x minus 1 plus the base 2 logarithm of x plus 1 equals 3 and gets x equal to 3 or negative 3. What is wrong?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | x = 13 | Rewrite as an exponential: x plus 3 is 16, so x is 13. |
| 2 | x = 13 | Rewrite as an exponential: 2x minus 1 is 25, so x is 13. |
| 3 | x = 0 or x = 2 | Let y be 2 to the power x; y squared minus 5y plus 4 gives y equal to 1 or 4, so x is 0 or 2. |
| 4 | x = 5 | Combining gives x times x minus 3 equal to 10, so x is 5 or negative 2, and the negative root is rejected. |
| 5 | 5 | The first two terms combine to the base 2 logarithm of 4, which is 2, and adding 3 gives 5. |
| 6 | 1.3010 | Since 20 is 4 times 5, the logarithm is twice 0.3010 plus 0.6990, which is 1.3010. |
| 7 | x = 3 | Writing both sides as powers of 3 gives 2x equal to 3x minus 3, so x is 3. |
| 8 | x = 6 | Writing both sides as powers of 2 gives 3x equal to 2x plus 6, so x is 6. |
| 9 | 3 hours | Growing from 500 to 4000 is an eightfold increase, and 2 cubed is 8, so it takes 3 hours. |
| 10 | Negative 3 is rejected, so the answer is only x = 3 | Combining gives x squared minus 1 equal to 8, but x equal to negative 3 makes a logarithm of a negative number, which is undefined. |
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.
Substitute every root back into a logarithm equation and write why it is kept or rejected.Tutor Registration →
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.