Delta method applied to $\log\bar X$ for a Gamma-distributed sample

Let $X_1,\dots,X_n$ be iid $f(x;\theta,\alpha)= \dfrac{e^{-x/\theta}x^{\alpha-1}}{\Gamma(\alpha)\theta^\alpha}$. Then, $\sqrt n(\log\bar X-\log\alpha\theta)\overset{d}{\longrightarrow} z$, where z is distributed as: