I am puzzled by why a moderator deleted my answer:
Here is the original answer in full:
Other answers address the question about the natural logarithm directly.
In more generality, why would you expect that for an arbitrary function $f,$ the mean value of $f(x)$ should be $f(0.5)$?
It doesn't answer the question. Indeed, it's explicitly framed as another question! That's ideal as a Socratic comment, but it definitely should not stay up as an answer.
Perhaps that post should have been converted to a comment, another option available at the time. I would be glad to do that retroactively.