I know that lots of divergences between two distributions are based on the expected value of one distribution with another one. Assume that for two distributions $p(x)$ and $q(x)$, we want to calculate the following equation:
$E_{p(x)}[log(q(x))] = \int_x p(x) log(q(x))$
But I want to know that what the intuition behind this expectation is. In a normal distribution, when we have random variable $X$ and a distribution on its outcomes ($p(x)$), the expectation actually is a weighted sum of the outcomes of $X$ weighted by their probability of occurrence. But in the above form, $q(x)$ is another distribution, so how can we interpret the meaning of the above expectation?