Express $y$ in terms of $x$ given ${log}_{2} y + {log}_{4} x + {log}_{4} 2 x = 0$ .

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Solution

$A] l o g_{2} y + l o g_{4} x + l o g_{4} 2 x = 0$
$= \frac{l o g y}{l o g 2} + \frac{l o g x}{l o g 2^{2}} + \frac{l o g 2 x}{l o g 2^{2}} = 0$
$= \frac{2 l o g y}{2 l o g 2} + \frac{l o g x}{2 l o g^{2} + \frac{l o g 2 x}{2 l o g^{2}}} = 0$
$= 2 l o g y + l o g x + l o g 2 x = 0$
$= 2 l o g y = - l o g (x \times 2 x)$
$= l o g y^{2} = l o g (2 x^{2})^{- 1}$
$= y^{2} = \frac{1}{2 x^{2}}$

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