Product of random variables is uniform

Question

Require a random variable $X$, $0 < X < 1$ with a distribution such that the product of two independent samples from the distribution is uniformly distributed on the interval (0,1).

Answer 1

If $U \sim \text{Uniform}(0,1)$ then observe that $-\ln(U) \sim \text{Exp}(1)$. Hence if $X, X'$ are i.i.d. such that $XX' \overset{d}{=} U$, then $-\ln(X)-\ln(X') \overset{d}{=} \text{Exp}(1)$.

But also, $\text{Exp}(1) = \text{Gamma}(1,1)$, and we know that the sum of independent Gamma distributions with the same rate parameter is a Gamma distribution with the sum of the shape parameters, so we deduce that we must have $-\ln(X) \sim \text{Gamma}(1/2,1)$.