Let $X$ be a random variable which is the difference of two binomial random variables, $$X=Y_1-Y_2,\\ Y_1\sim {\rm Binomial}(n,p_1),\\ Y_2\sim {\rm Binomial}(n,p_2).$$
Note that the first parameter of the binomial distribution is the same ($n$) for both $Y_1$ and $Y_2$. Is it possible to compute the distribution of $Z=\log(1 + e^X)$ ? If not the distribution, can we at least compute the expectation $\mathbb{E}[Z]$ or say anything about it?