| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Simplify or prove logarithmic identity |
| Difficulty | Moderate -0.8 Part (i) requires basic recall of fundamental logarithm definitions. Part (ii) is a straightforward algebraic manipulation using standard log laws (power rule, quotient rule, product rule) with no problem-solving insight needed—purely mechanical application of rules to verify an identity. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks |
|---|---|
| (i) \(\log_a 1 = 0\), \(\log_a a = 1\) | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \log_a x^{10} - \log_a \frac{x^6}{16}\) | M1 | Use of power law |
| \(= \log_a \frac{16x^{10}}{x^6} = \log_a 16x^4\) | M1 | Use of subtraction law |
| \(= \log_a (2x)^4 = 4\log_a(2x)\) | A1 | Completion |
## Question 6:
**(i)** $\log_a 1 = 0$, $\log_a a = 1$ | B1 B1 |
**(ii)** $\log_a x^{10} - 2\log_a\left(\frac{x^3}{4}\right)$
$= \log_a x^{10} - \log_a \frac{x^6}{16}$ | M1 | Use of power law
$= \log_a \frac{16x^{10}}{x^6} = \log_a 16x^4$ | M1 | Use of subtraction law
$= \log_a (2x)^4 = 4\log_a(2x)$ | A1 | Completion
---6 (i) Write down the values of $\log _ { a } 1$ and $\log _ { a } a$, where $a > 1$.\\
(ii) Show that $\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )$.
\hfill \mbox{\textit{OCR MEI C2 2007 Q6 [5]}}