smc

The surface area of this right rectangular prism is $2({\log }_{2}x{\log }_{3}x+{\log }_{2}x{\log }_{4}x+{\log }_{3}x{\log }_{4}x).$ The volume of this right rectangular prism is ${\log }_{2}x{\log }_{3}x{\log }_{4}x.$ Equating the numerical values of the surface area and the volume, we have $$2({\log }_{2}x{\log }_{3}x+{\log }_{2}x{\log }_{4}x+{\log }_{3}x{\log }_{4}x)={\log }_{2}x{\log }_{3}x{\log }_{4}x.$$ Dividing both sides by ${\log }_{2}x{\log }_{3}x{\log }_{4}x,$ we get $$2\left(\frac{1}{{\log }_{4}x}+\frac{1}{{\log }_{3}x}+\frac{1}{{\log }_{2}x}\right)=1.(★)$$ Recall that ${\log }_{b}a=\frac{1}{{\log }_{a}b}$ and ${\log }_b\left(a^n\right)=n{\log }_{b}a,$ so we rewrite $(★)$ as $$\begin{array}{rl}2({\log }_{x}4+{\log }_{x}3+{\log }_{x}2)& =1\\ 2{\log }_{x}24& =1\\ {\log }_{x}576& =1\\ x& =\boxed{576}.\end{array}$$