Find the value of \(p\) for which \(\sqrt { 125 } = 5 ^ { p }\).
Hence solve the equation \(5 ^ { 2 x } = \sqrt { 125 }\).
Use logarithms to solve the equation \(3 ^ { 2 x - 1 } = 0.05\), giving your value of \(x\) to four decimal places.
It is given that
$$\log _ { a } x = 2 \left( \log _ { a } 3 + \log _ { a } 2 \right) - 1$$
Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms.
(4 marks)