| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - linked parts (hence) |
| Difficulty | Moderate -0.8 This is a straightforward logarithm manipulation question testing standard log laws (product, quotient, power rules). Part (a) and (b) require direct application of basic rules with no problem-solving, while part (c) is a routine algebraic equation following from the previous parts. The question is easier than average A-level standard as it's purely procedural with clear scaffolding. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\log_2(16x) = \log_2 16 + \log_2 x = 4 + a\) | M1, A1 c.a.o | (2 marks) |
| (b) \(\log_2(\frac{x^4}{2}) = \log_2 x^4 - \log_2 2\) | M1 | |
| \(= 4\log_2 x - \log_2 2\) | M1 | |
| \(= 4a - 1\) (accept \(4\log_2 x - 1\)) | A1 | (3 marks) |
| (c) \(\frac{1}{2} = 4a - (4a - 1)\) | M1 | |
| \(a = \frac{3}{2}\) | A1 | |
| \(\log_2 x = \frac{3}{2} \Rightarrow x = 2^{\frac{3}{2}}\) | M1 | |
| \(x = \sqrt{8}\) or \(2\sqrt{2}\) or \(2^3\) or \((\sqrt{2})^3\) | A1 | (4 marks) |
**(a)** $\log_2(16x) = \log_2 16 + \log_2 x = 4 + a$ | M1, A1 c.a.o | (2 marks)
**(b)** $\log_2(\frac{x^4}{2}) = \log_2 x^4 - \log_2 2$ | M1 |
$= 4\log_2 x - \log_2 2$ | M1 |
$= 4a - 1$ (accept $4\log_2 x - 1$) | A1 | (3 marks)
**(c)** $\frac{1}{2} = 4a - (4a - 1)$ | M1 |
$a = \frac{3}{2}$ | A1 |
$\log_2 x = \frac{3}{2} \Rightarrow x = 2^{\frac{3}{2}}$ | M1 |
$x = \sqrt{8}$ or $2\sqrt{2}$ or $2^3$ or $(\sqrt{2})^3$ | A1 | (4 marks)Given that $\log_2 x = a$, find, in terms of $a$, the simplest form of
\begin{enumerate}[label=(\alph*)]
\item $\log_2 (16x)$, [2]
\item $\log_2 \left( \frac{x^4}{2} \right)$. [3]
\item Hence, or otherwise, solve $\log_2 (16x) - \log_2 \left( \frac{x^4}{2} \right) = \frac{1}{2}$, giving your answer in its simplest surd form. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q4 [9]}}