If $\ln(x)$ is gamma distributed, what is the distribution of $x$? Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.

Just do derive what gt6989b said:
$$
\frac{f_Y(\ln x)}{x} = \frac{\beta^\alpha}{\Gamma(\alpha)x}\ln(x)^{\alpha-1}e^{-\beta\ln(x)} = \frac{\beta^\alpha}{\Gamma(\alpha)}\ln(x)^{\alpha-1}e^{-\beta} 
$$
