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If $\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$, then what is $y$ equal
to?
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Verified Answer
The correct answer is:
9
$\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$
$\frac{\log x}{\log 3} \times \frac{\log 2 x}{\log x} \times \frac{\log y}{\log 2 x}=\frac{\log x^{2}}{\log x}$
$\frac{\log y}{\log 3}=\frac{2 \log x}{\log x}$
$\log y=2 \log 3$
$\log y=\log 9$
$y=9$
$\frac{\log x}{\log 3} \times \frac{\log 2 x}{\log x} \times \frac{\log y}{\log 2 x}=\frac{\log x^{2}}{\log x}$
$\frac{\log y}{\log 3}=\frac{2 \log x}{\log x}$
$\log y=2 \log 3$
$\log y=\log 9$
$y=9$
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