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If $\left(\log _{5} x\right)\left(\log _{x} 3 x\right)\left(\log _{3 x} y\right)=\log _{x} x^{3},$ then $y$ equals
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The correct answer is:
125
We have,
$\begin{aligned} & \log _{5} x \cdot \log _{x} 3 x \cdot \log _{3 x} y=\log _{x} x^{3} \\ \Rightarrow & \frac{\log x}{\log 5} \times \frac{\log 3 x}{\log x} \times \frac{\log y}{\log 3 x}=3 \log _{x} x \end{aligned}$
$\left[\because \log _{b} a=\frac{\log a}{\log b}\right.$ and $\left.\log a^{m}=m \log a\right]$
$\frac{\log y}{\log 5}=3$
$\left[\because \log _{a} a=1\right]$
$\begin{array}{ll}\Rightarrow & \log y=3 \log 5 \\ \Rightarrow & \log y=\log 5^{3}\end{array}$
$y=5^{3}=125$
$\begin{aligned} & \log _{5} x \cdot \log _{x} 3 x \cdot \log _{3 x} y=\log _{x} x^{3} \\ \Rightarrow & \frac{\log x}{\log 5} \times \frac{\log 3 x}{\log x} \times \frac{\log y}{\log 3 x}=3 \log _{x} x \end{aligned}$
$\left[\because \log _{b} a=\frac{\log a}{\log b}\right.$ and $\left.\log a^{m}=m \log a\right]$
$\frac{\log y}{\log 5}=3$
$\left[\because \log _{a} a=1\right]$
$\begin{array}{ll}\Rightarrow & \log y=3 \log 5 \\ \Rightarrow & \log y=\log 5^{3}\end{array}$
$y=5^{3}=125$
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