Question

If $\left({\log }_{5}x\right)\left({\log }_{x}3x\right)\left({\log }_{3x}y\right)={\log }_x{x}^3$, then y equals

• A. 125
• B. 25
• C. 5/3
• D. 243

Answer 1

Answer: (A)

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