Purpose: Construct a 95% confidence interval for $\theta$.
A common strategy is to construct a confidence interval for $\log(\theta)$ and then exponentiate.
Why is it valid?
My concern is that $E(\log(\theta)) \neq \log(E(\theta))$ and $Var(\log(\theta)) \neq \log(Var(\theta))$...
Do people use this back transformation because it has proven to work fine? What about other transformations (e.g. logit)?