The motivation of the value function of the generative adversarial network The seminal paper on the generative adversarial network, proposes to
$$\min_G\max_D V(D,G)$$
where the value function
$$V(D,G):=\int p_{\text{data}}(x)\ln p(D=\text{data}|x)\,dx+\int p_{\text{gen}}(x)\ln p(D=\text{gen}|x)\,dx.$$
Here I rewrite the expression $D(x)$ in the original paper in the format of a condition probability $p(D|x)$ to make it explicitly clear. My question is what motivates this particular expression? However it is not clear to me why the function the expectations operate on should be a logarithmic function. An alternative could be to simply drop the logarithmic function and use directly the probability $p(D=\text{data})$ for the data and $p(D=\text{gen})$ for the generated samples.

I will answer my own question. This is formulated in terms of conditional cross entropy. I will me expand on the details as follows.
