Answer:
yes, you can define logf even in much greater generality than in your situation.
Namely, if you have a real differentiable manifold M and a function f∈C∞(M) never zero on M, then f has a C∞ logarithm as soon as the first De Rham cohomology group of M vanishes: H1DR(M,R)=0.
The definition of the logarithm is straightforward: fix a point x0∈M and define
(logf)(x)=∫γdff
where γ is a differentiable path joining x0 to x, along which we can integrate the closed 1-form dgg.
The vanishing cohomology hypothesis ensures that the value of logf at x does not depend of the path γ chosen.
If M happens to be a complex holomorphic manifold and if f∈O(M) is holomorphic, then the logarithm of f will automatically be holomorphic: logf∈O(M).
This applies to your case since a simply connected manifold -and a fortiori a convex set in a vector space- has zero first De Rham cohomology group.
Finally, just for old times' sake, let me sum this up in the language of classical physics :
Every conservative vector field has a potential
A variant
Specialists in complex manifolds are addicted to the exact sequence of sheaves on the complex manifold X:
0→2iπZ→OX→expO∗X→0
A portion of the associated cohomology long exact sequence is
Γ(X,OX)→expΓ(X,O∗X)→H1(X,Z)
which shows again that every nowhere vanishing holomorphic function on X is an exponential (in other words: has a logarithm) as soon as H1(X,Z)=0.
