Example 2. If $\lg ^{2} x \lg 10 x<0$, find the value of $\frac{1}{\lg 10 x} \sqrt{\lg ^{2} x+\lg 10 x^{2}}$.
From the known condition $\lg ^{2} x \lg 10 x<0$, it is not difficult to conclude that $\lg x \neq 0$ and $\lg 10 x<0$, which also implicitly indicates the existence of $\lg x$.
$$
\text { Hence } \begin{aligned}
& \frac{1}{\lg 10 x} \sqrt{\lg ^{2} x+\lg 10 x^{2}} \\
= & \frac{1}{\lg 10 x} \sqrt{\lg ^{2} x+2 \lg x+\lg 10} \\
= & \frac{1}{\lg 10 x} \sqrt{(\lg x+1)^{2}} \\
= & \frac{1}{\lg 10 x} \sqrt{\lg ^{2} 10 x} \\
= & \frac{-\lg 10 x}{\lg 10 x}=-1 .
\end{aligned}
$$