## Task $3 / 78$

The equation $x^{\lg x}=100 x$ is to be solved for real numbers $x$!
From the given equation, by taking the logarithm of both sides and applying the logarithm laws, we get

$$
\begin{gathered}
\lg x^{\lg x}=\lg 100 x \rightarrow \lg x \cdot \lg x=\lg 100+\lg x \rightarrow \lg ^{2} x=\lg x+\lg 2 \rightarrow \\
\lg ^{2} x-\lg x-2=0
\end{gathered}
$$

This quadratic equation in $\lg x$ has the solutions

$$
\begin{array}{lll}
\lg x_{1}=0.5+\sqrt{0.25+2}=2 & ; & x_{1}=100 \\
\lg x_{2}=0.5-\sqrt{0.25+2}=-1 & ; & x_{2}=0.1
\end{array}
$$

Verification confirms the correctness.