A while ago I started reading these notes on special functions, but I got stuck trying to find all positive numbers $T$ satisfying $\int_T^\infty x^{-\log{x}} dx = \int_0^Tx^{-\log{x}}dx$
I noticed the integrand is $e^{-(\log{x})^2}$, but this isn't quite a Gaussian because the former is missing a factor of $x^{-1}$. Based on the straightforwardness of the other problems, I suspect there is a clever way of doing this, but unfortunately I'm not seeing it at the moment.
Hints would be greatly appreciated! Complete solutions are alright too, but in that case would you please use spoiler formatting? Thanks!