| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Simplify or prove logarithmic identity |
| Difficulty | Moderate -0.5 Part (i) is straightforward application of logarithm laws (ln(ab) = ln a + ln b, ln(a^n) = n ln a, ln(e) = 1) with simple arithmetic. Part (ii) requires taking logarithms of both sides and solving an inequality, which is slightly more involved but still routine C3 material. The given exponential forms make the algebra clean, and no novel insight is required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: Attempt use of at least one logarithm property correctly applied to \(\ln\left(\frac{ep^2}{q}\right)\) | M1 | Not including \(\ln e = 1\); such as \(... = \ln ep^2 - \ln q\) |
| Obtain 261 legitimately with necessary detail seen | A2 | AG; award A1 if nothing wrong but not quite enough detail, or one slip on way to 261 |
| OR: Express \(\frac{ep^2}{q}\) in form \(e^n\) | M1 | With correct treatment of powers |
| Obtain \(e^{261}\) and hence 261 | A2 | AG; award A1 if not quite enough detail to be fully convincing |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Introduce logarithms and bring power down | M1 | Relating \(n\ln 5\) to a constant; if using base 5 or base 10, powers must remain on right-hand side |
| Obtain \(n\ln 5 > 580\) | A1 | Or equiv such as \(n > 580\log_5 e\) or \(n\log 5 > 580\log e\); allow eqn at this stage |
| State single integer 361 | A1 | Not \(n > 360\) nor \(n \geq 361\) |
# Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** Attempt use of at least one logarithm property correctly applied to $\ln\left(\frac{ep^2}{q}\right)$ | M1 | Not including $\ln e = 1$; such as $... = \ln ep^2 - \ln q$ |
| Obtain 261 legitimately with necessary detail seen | A2 | AG; award A1 if nothing wrong but not quite enough detail, or one slip on way to 261 |
| **OR:** Express $\frac{ep^2}{q}$ in form $e^n$ | M1 | With correct treatment of powers |
| Obtain $e^{261}$ and hence 261 | A2 | AG; award A1 if not quite enough detail to be fully convincing |
---
# Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Introduce logarithms and bring power down | M1 | Relating $n\ln 5$ to a constant; if using base 5 or base 10, powers must remain on right-hand side |
| Obtain $n\ln 5 > 580$ | A1 | Or equiv such as $n > 580\log_5 e$ or $n\log 5 > 580\log e$; allow eqn at this stage |
| State single integer 361 | A1 | Not $n > 360$ nor $n \geq 361$ |
---2 It is given that $p = \mathrm { e } ^ { 280 }$ and $q = \mathrm { e } ^ { 300 }$.\\
(i) Use logarithm properties to show that $\ln \left( \frac { \mathrm { e } \mathrm { p } ^ { 2 } } { q } \right) = 261$.\\
(ii) Find the smallest integer $n$ which satisfies the inequality $5 ^ { n } > p q$.
\hfill \mbox{\textit{OCR C3 2012 Q2 [6]}}