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.