I have a discrete random variable $X$ with $P(X \geq x) = c^x$ and I would like to bound $E(\log{X})$. I can write this as follows I think $E(\log{X}) = \sum_{x=1}^{\infty} c^x \log{x}.$
We know that $0\leq c \leq 1$. I would like to bound $E(\log{X})$ above and below.
One would approach would be to replace the sum by an integral but I didn't get anywhere. Can anyone see how to get good bounds?
Question has been edited to make it clearer.