In particular, I have
$$u = e^{-\gamma c_{t}}$$
Is it true that ($c_{t} > 0 \ \gamma \in \mathbb{R})$,
$$\log(\mathbb{E}\left[e^{-\gamma c_{t}} \right]) = \mathbb{E}\left[\log(e^{-\gamma c_{t}}) \right]$$
where $\mathbb{E}$ denotes expectation.
So, I know that $u$ is convex, and I am pretty sure that from Jensen's inequality,
$$\log(\mathbb{E}\left[e^{-\gamma c_{t}} \right]) \geq \mathbb{E}\left[\log(e^{-\gamma c_{t}}) \right]$$
But it doesn't seem like I can produce a strict equality.
EDIT: Some additional info, $y_{t}$ is i.i.d.
$$c_{t} = \frac{r}{1+r}\left(A_{t}+y_{t}+\frac{1}{r}\bar{y} \right)-\pi(r,\gamma)$$
And, $y_{t} = \bar{y}-\epsilon_{t}$ and finally, $E_{t-1}[\epsilon_{t}] = 0$
This is part of a homework question, however, my question in particular is just something that would make a potential proof for the problem make much more sense (so I am not asking a solution to a homework problem, but help with a method I am using to solve a homework problem)