5.[In this question the values of $a , x$ ,and $n$ are such that $a$ and $x$ are positive real numbers,with $a > 1 , x \neq a , x \neq 1$ and $n$ is an integer with $n > 1$ ]

Sam was confused about the rules of logarithms and thought that

$$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$
\begin{enumerate}[label=(\alph*)]
\item Given that $x$ satisfies statement(1)find $x$ in terms of $a$ and $n$ .

Sam also thought that

$$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$
\item For $n = 3 , x _ { 1 }$ and $x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)$ are the two values of $x$ that satisfy statement(2).
\begin{enumerate}[label=(\roman*)]
\item Find,in terms of $a$ ,an expression for $x _ { 1 }$ and an expression for $x _ { 2 }$ .
\item Find the exact value of $\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)$ .
\end{enumerate}\item Show that if $\log _ { a } x$ satisfies statement(2)then

$$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2012 Q5 [14]}}