Solve the functional equation:
$\displaystyle f(xy)=[f(x)]^{y^\beta}[f(y)]^{x^\beta}$, where $\displaystyle x>0,~f(x)>0,~\beta\in\mathbb{R}$.
P.S. Good luck!
let b=beta
taking the logarithm of the both sides , we get :
$\displaystyle ln(f(xy))= y^bln(f(x))+x^bln(f(y))$ , let $\displaystyle g(x)=ln(f(x))$
we get : $\displaystyle g(xy)=y^bg(x)+x^bg(y)$
dividing both sides by $\displaystyle x^by^b$:
$\displaystyle \frac{g(xy)}{x^by^b} = \frac{g(x)}{x^b} + \frac{ g(y)}{y^b}$
let : $\displaystyle h(x)= e^{\frac{g(xy)}{x^by^b}} $
we have : $\displaystyle h(xy)=h(x)+h(y)$ which is equivalent to : $\displaystyle k(xy)=k(x)k(y)$ , if i remember well , the solution of the Last equation is $\displaystyle x^a$ for some real a , a difficult result of Erdos !!