| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 7 |
| 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 basic log laws (subtraction, power rules) and solving a linear equation in y. Part (a) involves routine application of log properties, while part (b) is a direct substitution leading to a simple linear equation. The question is easier than average as it requires only standard techniques with clear scaffolding and no problem-solving insight. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
\begin{enumerate}[label=(\alph*)]
\item Given that
$$y = \log_3 x$$
find expressions in terms of $y$ for
\begin{enumerate}[label=(\roman*)]
\item $\log_3\left(\frac{x}{9}\right)$
\item $\log_3 \sqrt{x}$
\end{enumerate}
Write each answer in its simplest form. [3]
\item Hence or otherwise solve
$$2\log_3\left(\frac{x}{9}\right) - \log_3 \sqrt{x} = 2$$ [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q5 [7]}}