| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve using substitution or auxiliary variable |
| Difficulty | Moderate -0.3 This is a straightforward C2 logarithm question requiring basic log laws (quotient, power, and root rules) and simple algebraic manipulation. Part (a) guides students through the substitution, making part (b) routine. The multi-step nature and substitution technique place it slightly below average difficulty, but it remains a standard textbook exercise with no novel insight required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(= \log_2 x - \log_2 2 = y - 1\) | M1 A1 | |
| (ii) \(= \log_2 x^3 = \frac{1}{2}\log_2 x = \frac{1}{2}y\) | M1 A1 | |
| (b) \(2(y-1) + \frac{1}{2}y = 8\) | M1 | |
| \(y = 4\) | ||
| \(\log_2 x = 4, \quad x = 2^4 = 16\) | M1 A1 | (7) |
**(a)** (i) $= \log_2 x - \log_2 2 = y - 1$ | M1 A1 |
(ii) $= \log_2 x^3 = \frac{1}{2}\log_2 x = \frac{1}{2}y$ | M1 A1 |
**(b)** $2(y-1) + \frac{1}{2}y = 8$ | M1 |
$y = 4$ | |
$\log_2 x = 4, \quad x = 2^4 = 16$ | M1 A1 | **(7)**3. (a) Given that $y = \log _ { 2 } x$, find expressions in terms of $y$ for
\begin{enumerate}[label=(\roman*)]
\item $\quad \log _ { 2 } \left( \frac { x } { 2 } \right)$,
\item $\log _ { 2 } ( \sqrt { x } )$.\\
(b) Hence, or otherwise, solve the equation
$$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q3 [7]}}