Question

If $\left({\log }_{3}x\right)\left({\log }_{x}2x\right)\left({\log }_{2x}y\right)={\log }_x{x}^2$, then what is $y$ equal to?

• A. $4.5$
• B. $9$
• C. $18$
• D. $27$

Answer 1

The correct option is B $9$

Given, $\left({\log }_{3}x\right)\left({\log }_{x}2x\right)\left({\log }_{2x}y\right)={\log }_x{x}^2$ .... $\left(i\right)$

Consider, $\left({\log }_{3}x\right)\left({\log }_{x}2x\right)\left({\log }_{2x}y\right)$

$=\left(\frac{\log x}{\log 3}\right)\left(\frac{\log 2x}{\log x}\right)\left(\frac{\log y}{\log 2x}\right)$

$=\frac{\log y}{\log 3}$

$={\log }_{3}y$
Also, ${\log }_x{x}^2=2{\log }_{x}x=2$

Thus, ${\log }_{3}y=2$

$\Rightarrow y=3^2=9$
Hence, option B is correct