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]