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.

For a positive response, the model $\log Y=\eta +\epsilon$ with $\epsilon \sim N(0,{\sigma }^2)$ gives conditional median $e^{\eta }$ and conditional mean $e^{\eta +{\sigma }^2/2}$. The latter follows from the moment-generating function of a normal distribution. Additive factor effects in $\eta$ become multiplicative effects on these response summaries. Its constant log-scale variance corresponds to original-scale variance proportional to the square of the mean; direct exponentiation of fitted log means incurs retransformation bias.

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 $e^{{\sigma }^2/2}$.