If $({log}_{3} x) ({log}_{x} 2 x) ({log}_{2 x} y) = {log}_{x} x^{2}$ , then $y$ equals:

\frac{9}{2}

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9

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18

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27

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81

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Solution

The correct option is B $9$
Since ${log}_{a} b = \frac{{log}_{c} b}{{log}_{c} a}$ , we have with base $x$ ,
$\frac{log x}{log 3} . \frac{log 2 x}{log x} . \frac{log y}{log 2 x} = 2; log y = 2 log 3 = log 9; y = 9$

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