smc

Taking the base-$x$ logarithm of both sides gives ${\log }_{10}x={\log }_x\frac{x^3}{100}$. This simplifies to $${\log }_{10}x={\log }_x{x}^3-{\log }_{x}100$$ $${\log }_{10}x+{\log }_{x}100=3$$ $${\log }_{10}x+2{\log }_{x}10=3$$ At this point, we substitute $u={\log }_{10}x$. $$u+\frac{2}{u}=3$$ $$u^2+2=3u$$ $$u^2-3u+2=0$$ $$(u-2)(u-1)=0$$ $$u\in \{1,2\}$$ $$x\in \{10,100\}$$ The answer is $\boxed{\text{10 or 100, only}}$.