Answer
$\{e,e^{4}\}$
Work Step by Step
For the equation to be defined, it must be true that
(*)$\left\{\begin{array}{ll}
x \gt 0 & \\
\ln x \geq 0 & \Rightarrow x \geq 1
\end{array}\right..\quad $Thus, $x \geq 1$
... Substituting $t=\sqrt{\ln x}$ , ($ t \geq 0$ ), we have
$t^{2}-3t+2=0$
$t=\displaystyle \frac{3\pm\sqrt{9-4(1)(2)}}{2}$
$t=\displaystyle \frac{3\pm 1}{2}$
$\left[\begin{array}{lll}
t=2 & or & t=1\\
\sqrt{\ln x}=2 & & \sqrt{\ln x}=1\\
\ln x=4 & & \ln x=1\\
x=e^{4} & or & x=e
\end{array}\right]$
both satisfy (*), so the solution set is
$\{e,e^{4}\}$
