E(log(x)) to E(x) Sorry if this is a straightforward question, but I have tried digging into econometrics book and cannot find anything about it. I worked on a model with log(wage) = experience + experience^2 + experience^3 and i have the results for that now. However, my results are in the form of E(log(wage)). Instead, now, I wanna move to E(wage).
Is there a transformation I should perform? Or does anyone have literature or terms I should google for?
Hopefully someone can help me
 A: The relationship between $E[\ln(x)]$ and $E[x]$ will in general depend on the distribution of $\ln(x)$.
If you fit $\ln(x)$ with an ordinary least squares (OLS) regression, then by assumption $\ln(x)$ is normally distributed around its expectation value with some standard deviation $\sigma_{\log}$, which should have been reported in by your OLS fitting software.  Therefore, $x$ is log-normal distributed.  Conveniently, the log-normal distribution is parameterized by the mean and standard deviation of the log, $\mu_{\log}$ and $\sigma_{\log}$.  In terms of these parameters, the expectation value of $x$ is $\exp\left(\mu_{\log} + \frac{\sigma^2_{\log}}{2}\right)$.  (The median, btw, is independent of $\sigma_{\log}$ in this case; it's just $\exp(\mu_{\log})$).
If you did something other than an OLS regression (e.g., you fit a GLM with a log link function), then the assumed conditional distribution for $\ln(x)$ is different, and therefore the relationship between the expectations of the log and the variable will be different.  In particular, for a Poisson regression, $E[x]$ is just $\exp(E[\ln(x)])$.  For other families you may have to look up the relationship.
Also, note that all of these formulae ignore the uncertainty in your estimate of $E[\ln(x)]$, which may or may not be a reasonable thing to do in your application.
