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If $\left(\log _{x} x\right)\left(\log _{3} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$, then what is the value
of $y$ ?
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of $y$ ?
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The correct answer is:
9
$\left(\log _{x} x\right)\left(\log _{3} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}$
$\left.\Rightarrow 1\left(\log _{3} 2 x\right)\left(\log _{2 x} y\right)=2\left(\because \log _{x} x^{2}\right)=2 \log _{x} x\right)$
$\Rightarrow\left(\frac{\log 2 x}{\log 3}\right)\left(\frac{\log y}{\log 2 x}\right)=2$
$\Rightarrow \frac{\log y}{\log 3}=2 \Rightarrow \log y=2 \log 3$
$\left.\Rightarrow 1\left(\log _{3} 2 x\right)\left(\log _{2 x} y\right)=2\left(\because \log _{x} x^{2}\right)=2 \log _{x} x\right)$
$\Rightarrow\left(\frac{\log 2 x}{\log 3}\right)\left(\frac{\log y}{\log 2 x}\right)=2$
$\Rightarrow \frac{\log y}{\log 3}=2 \Rightarrow \log y=2 \log 3$
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