I hope to show that for a binomial distribution $X\sim Bin(n,1/2)$, $\mathbb{E}\log(X+1) \ge \log(n/2)$. This reduces to an inequality: for any $n\ge 2$,
$$ \sum_{i=0}^n \log(1+i) \binom{n}{i}\left(\frac{1}{2}\right)^n \ge \log(n/2). $$
I checked this inequality with simulation and it's true for $n$ ranges from 2 to 10000. However, I have no idea how to prove this.