2023 AMC 12B Problem 7

Write $L=\log n.$ Then $\log(n^2)-(\log n)^2$ $=2L-L^2$ $=L(2-L),$ and the fraction is $\dfrac{L(2-L)}{L-3}.$ A sign chart shows this is $\ge 0$ exactly when $L\le 0$ or $2\le L\lt 3.$ Since $n$ is a positive integer, $L\le 0$ forces $n=1,$ while $2\le L\lt 3$ gives $100\le n\le 999,$ which is $900$ values. In total $1+900=901.$ Thus, the correct answer is E.