Solve the following system of equations

$$
y=10^{(\lg x)^{\lg x}} ;
$$

$$
(\lg x)^{\lg 2 x}=\lg y \cdot 10^{(\lg \lg x)^{2}}
$$
(1)-from

$$
\lg y=(\lg x)^{\lg x}
$$

substituting this value into (2)

$$
(\lg x)^{\lg x} \cdot(\lg x)^{\lg 2}=(\lg x)^{\lg x} \cdot 10^{(\lg \lg x) 2}
$$

$(\lg x)^{\lg x}$ can only be 0 if $\lg x=0$ (that is, if $x=1$), but then $(\lg x)^{\lg x}$ would be undefined. Therefore, the value $x=1$ must be excluded, and we can divide both sides of (3) by $(\lg x)^{\lg x}$:

$$
(\lg x)^{\lg 2}=10^{(\lg \lg x)^{2}}
$$

Since the right-hand side is positive, we can take the base 10 logarithm of both sides

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

(4) is satisfied first and foremost if

$$
\lg \lg x=0
$$

which means

$$
\lg x=1
$$

from which

$$
x_{1}=10, \quad y_{1}=10
$$

Assuming that $\lg \lg x \neq 0$, dividing (4) by $\lg \lg x$ gives

$$
\lg 2=\lg \lg x
$$

which means

$$
\lg x=2
$$

from which

$$
x_{2}=10^{2}=100, \quad y_{2}=10^{2^{2}}=10^{4}=10000
$$

By substitution, we can verify that the roots we have found indeed satisfy our system of equations.

Erzsébet Lukács (Nagykanizsa, chemical engineering technology II. year)