I am working through some practice problems before an exam, but I am getting stuck. The PDF is $f(x \mid \theta) = \theta x^{\theta - 1}$, where $0 < x < 1$.
I have already found the MLE as $$\hat{\theta}_{\text{MLE}}=\frac{-n}{\sum_{i = 1}^{n}\log{x_i}}$$
I've also shown that this is both a complete and sufficient statistic, so all I need to do is show that it's unbiased to show that its the UMVUE. This is where I'm drawing a blank. How do I show that it's unbiased?