Originally Posted by

**thardan** Hello everyone,

the title essentially sums it up: let $\displaystyle X\sim\mathcal{N}\left(\mu,\sigma\right)$. My questions are:

1. Is there an analytical result for $\displaystyle \mathrm{E}\left(\ln\left(X\right)\right)$?

2. If not, are there any appropriate approximations?

This question looks so innocent, I expect it must have been treated somewhere. I could, however, not find it in the relevant literature so far.

Note that

$\displaystyle E(\ln(X)) \propto \int \ln(x) \exp\left(- \frac{\left(x-\mu\right)^2}{\sigma^2}\right) \mathrm{d}x$

The variable transformation $\displaystyle Y:=\ln\left(X\right)$ leads to (ignoring integration bounds)

$\displaystyle E(Y) \propto \int y \underbrace{\exp\left(- \frac{\left(e^y-\mu\right)^2}{\sigma^2}\right)}_{p_Y\left(y\right) }\mathrm{d}y$

So, my questions could also be interpreted as for the expectation of the random variable $\displaystyle Y$ that is distributed according to a pdf that is proportional to $\displaystyle p_Y\left(y\right)$. Unfortunately, I could not find any results for such a distribution, too.

Any help is greatly appreciated!

Thardan