| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Solve by showing reduces to polynomial |
| Difficulty | Standard +0.3 This is a standard logarithm manipulation question requiring systematic application of log laws (power rule, addition rule) to reduce to a polynomial, followed by routine factorization given a root. The steps are predictable and well-practiced, making it slightly easier than average despite requiring careful algebraic manipulation. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
\begin{enumerate}
\item In this question you must show detailed reasoning.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.\\
(a) Given that
$$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$
show that
$$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
(b) Given also that - 1 is a root of the equation
$$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
solve
$$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$
\section*{(Total for Question 10 is 6 marks)}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q10 [6]}}