Question

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

• A. 125
• B. 25
• C. 5/3
• D. 243

Answer 1

Answer: (A)

We have,
$\log _{5} x \cdot \log _{x} 3 x \cdot \log _{3 x} y=\log _{x} x^{3} $
$\Rightarrow \frac{\log x}{\log 5} \times \frac{\log 3 x}{\log x} \times \frac{\log y}{\log 3 x}=3 \log _{x} x$
$\left[\because \log _{b} a=\frac{\log a}{\log b}\right.$ and $\left.\log a^{m}=m \log a\right]$
$\Rightarrow \frac{\log y}{\log 5}=3 $
$\left[\because \log _{a} a=1\right]$
$\Rightarrow \frac{\log y}{\log 5}=3$
$ \Rightarrow \log y =3 \log 5 $
$ \Rightarrow \log y =\log 5^{3}$
$\Rightarrow y=5^{3}=125$