Logarithmic transformation

A logarithmic transformation replaces a positive response $Y$ by $\log Y$, often converting multiplicative effects into additive effects and stabilizing relative rather than absolute variation.

In general, $\mathbb{E}[Y\mid X]$ is not obtained by exponentiating $\mathbb{E}[\log Y\mid X]$. For log-normal errors of variance ${\sigma }^2$, the correction factor is $e^{{\sigma }^2/2}$.