I am following This paper. In section 3.1. Justification of regularizer I am trying to duplicate their transition from $D(x)=\frac{p_x(x)}{p_x(x)+p_G(x)}$ to the form of $\log(p_X(x))= \log(D(x)) − \log(1 − D(x)) + \log(p_Z(z)) + \log(|\frac{\partial Z}{\partial x}|)$.
I am aware that the when the GAN converges $p_x=p_G$ hence $D(x)=0.5$ though I am not sure if it helps.
So far I got:
$$\log(D(x))=\log(p_x(x))-\log(p_x(x)+p_G(x))$$ $$\log(D(x))=\log(p_x(x))-\log(p_x(x)+p_z(z)|\frac{\partial Z}{\partial x}|)$$
How do I proceed?
The log of a sum is an indicator that your first equation won't be helpful, because you won't be able to simplify it. Replace $p_G(x)$ by $p_Z(z) \left| \frac{\partial z}{\partial x}\right|$ in the original equation, multiply both sides by the denominator, and rearrange to group the $p_X(x)$s together: $$ p_X(x)(1-D(x)) = D(x) p_Z(z) \left| \frac{\partial z}{\partial x}\right|.$$ Now take logs to get the result you want.