| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | November |
| Marks | 9 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - linked parts (hence) |
| Difficulty | Moderate -0.8 This is a straightforward logarithm manipulation question requiring only standard log laws (product, quotient, power rules). Parts (a) and (b) are direct applications of basic rules, while part (c) involves simple algebraic manipulation to solve a quadratic equation. The 'hence' structure guides students through the solution, and no novel insight is required—just routine application of A-level techniques. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
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}
(Total 9 marks)
\hfill \mbox{\textit{SPS SPS SM 2025 Q4 [9]}}