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

Question

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

Honestly, I have no idea. This is my first encounter with functional equations. — Max, Mar 20 '13 at 07:17

score 5 · Answer 1 · answered Mar 20 '13 at 07:19

5

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

answered Mar 20 '13 at 07:19

If x=1, I will get this function m(1)=m(1)+m(y), which implies m(y)=0? – Max Mar 20 '13 at 07:20
@Max Yes ${}{}{}{}{}$ – Mar 20 '13 at 07:20
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 07:24
@Max Yes ${}{}$ – Mar 20 '13 at 07:25

lsp · Answer 2 · 2014-02-12T11:46:51.690

1

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 !

edited Feb 12 '14 at 11:46

answered Mar 20 '13 at 08:13

lsp

2 Answers2