smc

From the properties of logarithms, we can simplify the equation and solve for $y$: $$\begin{array}{rl}({\log }_{3}x)({\log }_{x}2x)({\log }_{2x}y)& ={\log }_x{x}^2\\ ({\log }_{3}2x)({\log }_{2x}y)& =2{\log }_{x}x\\ {\log }_{3}y& =2\\ y& =3^2\\ y& =9\end{array}$$ Thus, our answer is $\boxed{9}$.