smc

We have $$\begin{array}{rl}\sqrt{\frac{\log \left(n^2\right)-{\left(\log n\right)}^2}{\log n-3}}& =\sqrt{\frac{2\log n-{\left(\log n\right)}^2}{\log n-3}}\\ & =\sqrt{\frac{\left(\log n\right)\left(2-\log n\right)}{\log n-3}}.\end{array}$$ Because $n$ is an integer and $\log n$ is well defined, $n$ must be a positive integer. Case 1: $n=1$ or $10^2$. The above expression is 0. So these are valid solutions. Case 2: $n\ne 1,10^2$. Thus, $\log n>0$ and $2-\log n\ne 0$. To make the above expression real, we must have $2<\log n<3$. Thus, $100<n<1000$. Thus, $101\le n\le 999$. Hence, the number of solutions in this case is 899. Putting all cases together, the total number of solutions is $\boxed{901}$.