How to log linearize an equation that has a conditional expectation? Any reference or example would be appreciated.
You need to remember two properties of derivatives and expectations. Everything else stays the same.
$E(f(x))$ is not equal to $f(E(x))$ unless f is linear.
$\partial E(f(x)) / \partial x$ is equal to $E(\partial f(x) / \partial x)$.
Also remember that as a result, linearizing models with uncertainty reduces them to their certainty equivalent version. This may or may not matter for what you are doing.