Question

Find all differentiable functions defined for x>0 which satisfy ƒ(xy)=ƒ(x)+ƒ(y), x,y>0.

Answer 1

logs work, but it says "find all" so we need a proof of some kind

Answer 2

just a couple quick thoughts. if
\[f(xy)=f(x)+f(y)\] then
\[f(x^n)=nf(x)\] for sure, and also
\[f(1)=0\] (from the previous consideration)

Answer 3

Is there a name/type of function that we could use to prove this?

Answer 4

Any idea, PsychoTink?

Answer 5

lol