$\ln(2e)^2 = 2\ln(2e)$ is not correct.

What is correct is

$\ln((2e)^2)=2\ln(2e)$

Looking at your problem.

$f(x)=(\ln(x))^2-3\ln(x)$

$f(2e)=(\ln(2e))^2 - 3\ln(2e)$

this is actually in fairly simple form. You could further do the following

$u=\ln(2e)$

$f(2e)=u^2-3u = (u-1.5)^2-2.25 = (\ln(2e)-1.5)^2-2.25$