| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | September |
| Marks | 8 |
| Topic | Laws of Logarithms |
| Type | Express log in terms of given variables |
| Difficulty | Standard +0.3 Part (a) requires systematic application of log laws (change of base, powers, roots) but follows a standard template. Part (b) involves combining logs, forming a quadratic, and checking validity of solutions - all routine A-level techniques with no novel insight required. Slightly easier than average due to straightforward algebraic manipulation. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules |
11.
\begin{enumerate}[label=(\alph*)]
\item Given that $\log _ { 3 } c = m$ and $\log _ { 27 } d = n$, express $\frac { \sqrt { c } } { d ^ { 2 } }$ in the form $3 ^ { y }$, where $y$ is an expression in terms of $m$ and $n$.
\item Show that the equation
$$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$
has only one solution and state its value.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q11 [8]}}