Let f be a continuous function with domain x>0 and let F be the function given by

for x>0. Suppose that F(ab) = F(a) + F(b) for all a>0 and b>0 and that F'(1) = 3.

a) Find f(1)

b) Prove aF '(ax) = F '(x) for every positive constant a

c) Find f(x), justify answer

a) is easy, it is zero because going from 1 to 1 is like going nowhere.

I don't understand this remark: F(x) is a primitive of f(x) and thus . Perhaps you confused between f(1) and F(1)?
b) is where I am having a little difficulty with. I am not sure how to solve this. I guess I use liebniz's rule for the original integral pf F(x)?