What property of integration over probability densities allows this? Here is the concerning part of the equation that I am trying to understand dealing with the variational lower bound and Kullback-Leibler divergence.
$$
\begin{aligned}
&- \int_Z q(Z) \log \frac{p(Z|X)}{q(Z)} \\
&= -\Big( \int_Z q(Z) \log \frac{p(X,Z)}{q(Z)} - \int_Z q(Z) \log p(X) \Big)
\end{aligned}
$$
I just don't get what property of integration would give this. $p(X, Z)$ would be equal to $p(X|Z)p(Z)$ by bayes theorem, so it must be doing that somewhat in reverse, but I can't justify why that would be

$P(Z|X)=\frac{P(Z,X)}{P(X)}=\frac{P(X,Z)}{P(X)}$ since $P(X,Z)=P(Z,X)$.
Edit
By the above equality, you have
$$\int_Z q(Z)\log\frac{p(Z|X)}{q(Z)}= \int_Z q(Z) \log \frac{p(X,Z)}{q(Z)p(X)} =\int_Z q(Z)\left[\log\frac{p(X,Z)}{q(Z)}-\log p(X)\right]$$
