8. The set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{4}} 10<-\frac{1}{2}\right., x \in \mathbf{N}^{*}\right\}$ has how many proper subsets?
8. $2^{90}-1$ is derived from $-1 \leqslant \log \frac{1}{1} 10\frac{1}{2}\left(x \in \mathbf{N}^{*}\right)$, hence $100>x \geqslant 10$, so the number of elements in the set is 90, and the number of its proper subsets is $2^{90}-1$.