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

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

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Do you have any ideas? What happens if $y=1$? Or if $x=1$? –  Henry Mar 20 '13 at 7:16
Honestly, I have no idea. This is my first encounter with functional equations. –  Max Mar 20 '13 at 7:17

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

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If x=1, I will get this function m(1)=m(1)+m(y), which implies m(y)=0? –  Max Mar 20 '13 at 7:20
@Max Yes ${}{}{}{}{}$ –  user17762 Mar 20 '13 at 7:20
So, this implies m(x)=0=m(y) for any y,x in R+. So, the only solution is m(x)=0? –  Max Mar 20 '13 at 7:24
@Max Yes ${}{}$ –  user17762 Mar 20 '13 at 7:25

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 !

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