A question about the logarithm of an arbitrary function

1

Is in general the log of an arbitrary function a convex function?
If yes, give me explicit results or any paper or book that answer this question.

Thanks,
Farzali

2

No. log(x) is concave, and is not convex. That is a rather simple counterexample.

log(f(x)) is concave if f(x) is concave.