Solve the following equation

$$
\sqrt{x^{\lg \sqrt{x}}}=10
$$
Raising both sides to the power of four

$$
x^{\lg \sqrt{x}}=10^{2}
$$

Taking the base 10 logarithm of both sides

$$
\lg \sqrt{x} \cdot \lg x=2
$$

that is

$$
\frac{1}{2} \lg x \cdot \lg x=2
$$

from which

$$
(\lg x)^{2}=4
$$

that is

$$
\lg x= \pm 2
$$

and thus

$$
x_{1}=10^{2}=100, \quad x_{2}=10^{-2}=\frac{1}{100}
$$

By substitution, we can verify that both values satisfy the equation.

Péntek László (Kunszentmiklós, Damjanich g. III. o. t.)