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 $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$ （a）Given that \(x\) satisfies statement（1）find \(x\) in terms of \(a\) and \(n\) ． Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$ （b）For \(n = 3 , x _ { 1 }\) and \(x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)\) are the two values of \(x\) that satisfy statement（2）．
（i）Find，in terms of \(a\) ，an expression for \(x _ { 1 }\) and an expression for \(x _ { 2 }\) ．
（ii）Find the exact value of \(\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)\) ．
（c）Show that if \(\log _ { a } x\) satisfies statement（2）then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$