1. The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}$ is
$$
2^{90}-1
$$
1.【Analysis and Solution】First, examine the number of elements in the set. For $x \in \mathbf{N}$
$$
\begin{array}{l}
-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2} \\
\Leftrightarrow-2<\lg \frac{1}{x} \leqslant-1 \\
\Leftrightarrow 1 \leqslant \lg x<2 \\
\Leftrightarrow 10 \leqslant x<100 .
\end{array}
$$

Thus, the size of the set is 90, and the number of proper subsets is $2^{90}-1$.