Function

[yo79]

Find all odd functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which has the following proprietes:
1)$f(x)>0, \forall x\in (0, \infty)$
2)$f(log_x y)+f(log_y x)=log_x f(y)+log_y f(x)$,$ \forall x,y \in (0, \infty ) - \{1 \}, x \neq y$

[*Mahi*]

Find all odd functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which has the following proprietes:
1)$f(x)>0, \forall x\in (0, \infty)$
2)$f(log_x y)+f(log_y x)=log_x f(y)+log_y f(x)$,$ \forall x,y \in (0, \infty ) - \{1 \}, x \neq y$

Setting $x=y$ in (2)
$2 f(log_x x) = 2 log_x f(x)$
$\Rightarrow f(1) = log_x f(x)$
$\Rightarrow x^ {f(1)} = f(x)$
So $f(x) = x^c$ with $f(1) = c$
Setting this in $(2)$
$(log_x y)^c+(log_y x)^c=log_x y^c+log_y x^c$
$\Rightarrow (log_x y)^c+(log_y x)^c=c(log_x y+log_y x)$
$\Rightarrow (log_x y)^c+\frac 1{log_x y}^c=c(log_x y+\frac 1{log_x y})$
Which holds only for $c= 1$.
So, the solution is $f(x) = x \forall x \in (0. \infty)$
As $f$ is odd, $f(x) = x \forall x \in \mathbb R$

[yo79]

2)$f(log_x y)+f(log_y x)=log_x f(y)+log_y f(x)$,$ \forall x,y \in (0, \infty ) - \{1 \},$ $x \neq y$

[*Mahi*]

2)$f(log_x y)+f(log_y x)=log_x f(y)+log_y f(x)$,$ \forall x,y \in (0, \infty ) - \{1 \},$ $x \neq y$

Aah the old problem $\neq$ was not rendering properly in my browser, sorry.