If $\ln(x)$ is gamma distributed, what is the distribution of $x$?

Question

Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.

Answer 1

HINT Let $\ln X = Y \sim \Gamma(\alpha, \beta)$ then $$ F_X(x) = \mathbb{P}[X \le x] = \mathbb{P}[\ln X \le \ln x] = F_Y(\ln x) $$ and using the Chain Rule, $$ f_X(x) = F'_X(x) = f_Y(\ln x)/x $$

Answer 2

Just do derive what gt6989b said:

$$ \frac{f_Y(\ln x)}{x} = \frac{\beta^\alpha}{\Gamma(\alpha)x}\ln(x)^{\alpha-1}e^{-\beta\ln(x)} = \frac{\beta^\alpha}{\Gamma(\alpha)}\ln(x)^{\alpha-1}e^{-\beta} $$