State the set of values of \(a\) for which the function \(x \mapsto {\log _a}x\) exists, for all \(x \in {\mathbb{R}^ + }\).

Given that \({\log _x}y = 4{\log _y}x\), find all the possible expressions of \(y\) as a function of \(x\).

\(a > 0\)     A1

\(a \ne 0\)     A1

[2 marks]

METHOD 1

\({\log _x}y = \frac{{\ln y}}{{\ln x}}\) and \({\log _y}x = \frac{{\ln x}}{{\ln y}}\)     M1A1

Note:     Use of any base is permissible here, not just “e”.

\({\left( {\frac{{\ln y}}{{\ln x}}} \right)^2} = 4\)     A1

\(\ln y =  \pm 2\ln x\)     A1

\(y = {x^2}\;\;\;\)or\(\;\;\;\frac{1}{{{x^2}}}\)     A1A1

METHOD 2

\({\log _y}x = \frac{{{{\log }_x}x}}{{{{\log }_x}y}} = \frac{1}{{{{\log }_x}y}}\)     M1A1

\({({\log _x}y)^2} = 4\)     A1

\({\log _x}y =  \pm 2\)     A1

\(y = {x^2}\;\;\;\)or\(\;\;\;y = \frac{1}{{{x^2}}}\)     A1A1

Note: The final two A marks are independent of the one coming before.

[6 marks]

Total [8 marks]