How do governments use propaganda to shape people’s ideas about their country and the outside world? No links please, will mark brainliest!

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.

b.

one company controls all the means of production of a product.

~coco