| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - simplify/express then solve |
| Difficulty | Moderate -0.8 This question tests basic logarithm laws (power rule and addition rule) and simple manipulation. Part (a) requires straightforward application of log rules with no problem-solving, while part (b) involves recognizing that y = 3^(3/2) and applying the definition of logarithms. All parts are routine recall and mechanical application of standard techniques, making this easier than average for A-level. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\log_2 x\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_2(8x^2) = \log_2 8 + \log_2 x^2\) | M1 | For relevant sum of logarithms |
| M1 | For relevant use of \(8 = 2^3\) | |
| A1 | 3 | For correct simplified answer |
| \(= 3 + 2\log_2 x\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\log_3 y = \log_3 27\) | M1 | For taking logs of both sides of the equation |
| A1 | For any correct expression for \(\log_3 y\) | |
| A1 | 3 | For correct simplified answer |
| Hence \(\log_3 y = \frac{3}{2}\) | ||
| Total: 7 |
## Part (a)
### (i)
$2\log_2 x$ | B1 | 1 | For correct answer
### (ii)
$\log_2(8x^2) = \log_2 8 + \log_2 x^2$ | M1 | For relevant sum of logarithms
| M1 | For relevant use of $8 = 2^3$
| A1 | 3 | For correct simplified answer
$= 3 + 2\log_2 x$ | |
## Part (b)
$2\log_3 y = \log_3 27$ | M1 | For taking logs of both sides of the equation
| A1 | For any correct expression for $\log_3 y$
| A1 | 3 | For correct simplified answer
Hence $\log_3 y = \frac{3}{2}$ | |
| **Total: 7** | |\begin{enumerate}[label=(\alph*)]
\item Express each of the following in terms of $\log_2 x$:
\begin{enumerate}[label=(\roman*)]
\item $\log_2(x^2)$, [1]
\item $\log_2(8x^2)$. [3]
\end{enumerate}
\item Given that $y^2 = 27$, find the value of $\log_3 y$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q3 [7]}}