Functional equation $m(x^y)=m(x)+m(y)$.

Question

Find all functions $m : \mathbb{R}^+ \to \mathbb{R}^+$ such that$$m(x^y)=m(x)+m(y)$$

Answer 1

Set $x=1$ and conclude what the function should be.

Answer 2

Since the function is true for all positive real values, it should obviously be true for $x=1$.

Substituting $x=1$, we get:

$m(1) = m(1) + m(y) \implies m(y) = 0$

So the function equation becomes : $m(x^y) = m(x)$

You can easily find the function value from here.

Hope the answer is clear !