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S4 Maths: Logarithms Paper 3 (Applications and Comparisons)
S4 Maths: Logarithms Paper 3 (Applications and Comparisons)
This paper puts logarithms into pH, earthquake magnitude, decibels and compound interest, then compares logs of different bases.
Quick answer 1. 3 | 2. 10^-4 mol/L | 3. 3 units | 4. 10 steps | 5. three | 6. about 9 years | 7. 30 dB | 8. 100 times and 1000 times | 9. log2 5 is larger | 10. 20 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 | Challenge |
02 Questions (10 items)
- FoundationA solution has hydrogen ion concentration 10^-3 mol/L. Find its pH.
- FoundationA solution has pH 4. Find its hydrogen ion concentration.
- FoundationTwo earthquakes differ by a factor of 1 000 in amplitude. How many units apart are their Richter magnitudes?
- FoundationA routine doubles its operations each step. How many steps take it from 1 to 1 024?
- IntermediateA radioactive sample falls to one eighth of its original amount. How many half-lives have passed?
- IntermediateAt 8 per cent a year compounded annually, how long does a sum take to double? (log10 2 about 0.3010, log10 1.08 about 0.0334)
- IntermediateSound intensity rises 1 000-fold. How many decibels is that increase?
- ChallengeEarthquake A has magnitude 7 and B has magnitude 5. Find the amplitude ratio and the energy ratio.
- ChallengeCompare log2 5 with log3 5 and give a reason.
- Error redoA student sees intensity 100 times greater and answers that loudness rises by 100 dB. What is correct?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | pH = -log10 (10^-3) = 3. | Each drop of 1 in pH means ten times the acidity. |
| 2 | The concentration is 10^-4 mol/L. | Reverse the definition of pH: ten to the power of minus pH. |
| 3 | M = log10 1000 = 3, so they are 3 units apart. | Magnitude is the common log of amplitude, so one unit means ten times the amplitude. |
| 4 | log2 1024 = 10, so it takes 10 steps. | Doubling problems are handled directly by base-2 logarithms. |
| 5 | (1/2)^n = 1/8 gives n = log2 8 = 3, that is three half-lives. | Half-lives use base 2; the denominator of the remaining fraction is a power of two. |
| 6 | n = 0.3010 / 0.0334 = 9.01, so about 9 years. | The rule of 72 estimates doubling time: 72 / 8 = 9. |
| 7 | 10 log10 1000 = 10 x 3 = 30, so the increase is 30 dB. | Decibels are ten times the common log, so tenfold intensity adds only 10 dB. |
| 8 | Amplitude ratio = 10^(7-5) = 100; energy ratio is about 10^(1.5 x 2) = 1000. | Energy scales with amplitude to the power 1.5, so the energy gap widens faster. |
| 9 | log2 5 is about 2.32 and log3 5 about 1.46, so log2 5 is larger. | For numbers above 1, a smaller base gives a larger logarithm. |
| 10 | 10 log10 100 = 20, so the rise is 20 dB. | Decibels are logarithmic, so a hundredfold intensity is only 20 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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