I am having difficult time figuring the answer to the following problem.
Let $X_i \sim Beta(1, \alpha) $ for some $i = 1, \ldots , n$ and $\alpha > 0$. Find the following expectation: $$ E\left [ \log (1 - X_1) \Bigg \vert \prod_1^n (1 - X_i) \right]$$
I have tried writing this out as an integral, but I quickly get stuck with complicated expressions. Is there an easier way to see the result?