I tried to prove (by contradiction) that, given a positive function $f(x)$, it is not $O(\log(f(x))$ (in this case, log denotes the base 2 logarithm).
The problem is that I do not know anything about the behaviour of the function (and therefore nothing about its log), so I can't really know how to proceed.
I have the same difficulties when trying to prove that $f(x)$ is not $O(f(x)-x)$.
I might even be wrong, but I have hardly an idea how to start proving/disproving these thins, and some help is more than appreciated.