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Function
Posted: Wed Mar 06, 2013 1:49 am
by 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$
Re: Function
Posted: Wed Mar 06, 2013 10:07 am
by *Mahi*
yo79 wrote: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, $f(x) = x \forall x \in \mathbb R$
Re: Function
Posted: Wed Mar 06, 2013 1:23 pm
by yo79
yo79 wrote:
2)$f(log_x y)+f(log_y x)=log_x f(y)+log_y f(x)$,$ \forall x,y \in (0, \infty ) - \{1 \},$ $x \neq y$
Re: Function
Posted: Wed Mar 06, 2013 7:06 pm
by *Mahi*
yo79 wrote:yo79 wrote:
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.