Where $X_{1}, X_{2}, \dots X_{n}$ is an iid distribution with pdf given by:
\begin{cases} \frac{1}{\theta}x^{1-\theta} \qquad &\text{If $0 \leq x \leq 1$} \\[5 pt] 0 \qquad &Otherwise \end{cases}
It can be shown using the M.L.E for estimator inference that $\theta$ is given by: $$\theta = \frac{-1}{n} \sum_{i = 1}^{n} \ln(X_{i})$$
To take the expectation of this estimator, what's the next step? I believe that I need to find the underlying distribution for $\theta$, correct?
I ask that question, only because a solution does not find this underlying distribution (or else is not explicit):
And what if I find $\theta$ to be $(1 - \bar{X})/\bar{X}$?
In that case, how would I find the underlying pmf of $\theta$?