| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 Part (i) is a straightforward logarithm equation requiring basic log laws and solving a linear equation. Part (ii) requires recognizing summation notation, applying log laws (log(y^r) = r·log(y)), and solving a quadratic, but the steps are mechanical once set up. This is slightly easier than average as it's primarily procedural with no novel insight required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules4.06b Method of differences: telescoping series |
\begin{enumerate}
\item (i) Using the laws of logarithms, solve
\end{enumerate}
$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$
(ii) Given that
$$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } ( y ) \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$
find $y$ in terms of $a$, giving your answer in simplest form.\\
(3)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q7 [6]}}