Wolfram says $(\ln 2)^2$ is transcendental. I think it says numbers of the form $(\ln a)^b$ are all transcendental, at least for integer $a$ and $b$, I didn't check further.
Maybe there is some corollary from Lindemann's theorem that says something about my question or powers of $\log'$s.
I searched briefly on google for some literature on the irrationality/transcendence on powers of logarithms, either papers or forums, but didn't find anything. Any help would be appreciated.
Suppose $(\ln a)^b$ is algebraic. There exists a nonzero polynomial $p(x)\in\mathbb Q[x]$ such that $p\left((\ln a)^b\right)=0$. Let $q(x)=p(x^b)\in\mathbb Q[x]$. Then, $q$ is nonzero and $q(\ln a)=0$. So, $\ln a$ is algebraic.
Now if $a$ is a positive algebraic number besides $1$, then it follows from the Lindemann-Weierstrass theorem that $\ln a$ is transcendental. We conclude that $(\ln 2)^2$ is transcendental.