Suppose $X\sim \mathrm{Binomial}(n,p),$ i.e. $X$ takes values in $\{0,1,2,\ldots, n\}$ and $P(X=i) = {n\choose i} p^i(1-p)^{n-i}.$

I am looking for a good estimate for $\mathbb{E}\log(X+\alpha).$ I am thinking of $\alpha$ as some fixed number, for example, $\alpha = 1,$ $p$ as a fixed number between 0 and 1, eg. $p=\frac{1}{3}$ while $n$ is some very large number. The kind of estimate I need may be clarified by the following aim:

Define $f(n): = \log(n+2\alpha) +p\mathbb{E}\log\frac{p}{X+\alpha} + (1-p)\mathbb{E}\log\frac{1-p}{n-X+\alpha}.$ I want to show that $f(n)\sim \frac{c}{n} + o\left(\frac{1}{n}\right)$ and I want to identify the constant $c.$