Answer
$$ \{ 1, e^2 \}$$
Work Step by Step
$$(\ln x)^2= \ln x^2 \quad \Rightarrow \quad (\ln x )(\ln x)= 2 \ln x \quad \Rightarrow \quad \ln x =0 \quad \text{ or } \quad \ln x=2 \quad \Rightarrow \quad x=1 \quad \text{ or } \quad x=e^2$$Both values satisfy the original equation:$$(\ln 1)^2 =0 = \ln 1^2, \\ (\ln e^2)^2=4=\ln (e^2)^2$$
