Take the 2-minute tour ×  Suppose that $f:\mathbb{R}\rightarrow
(0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two
functions and satisfies the following relation \begin{equation}
f(xy)=f(y)^{g(x)} \end{equation} for all $x,y\in \mathbb{R}$.
Can we determine the all solutions of this funtional equation
under some suitable conditions.  share|improve this question
Note that its easy to show that the function $g$ is a
multiplicative map i.e; $g(xy)=g(x)g(y)$.. –  Alisad Nov 11 '12
at 1:30 You can also show that there is some constant $C$ with
$\frac{g(x)}{\text{log}f(x)}=C$ whenever $\text{log}f(x)\ne 0$,
so $g(x)=C\text{log}f(x)$. This gives us the relation
$\text{log}f(xy)=C\text{log}f(x)\text{log}f(y)$, so either
$f(x)$ is constant or $f(0)=e$ and $f(1)=e^{1/C}$. –  rayradjr
Nov 11 '12 at 2:08 Thanks, by your idea the problem solve, since
the general solution of the multiplicative functional equation
studied by some athurs. –  Alisad Nov 11 '12 at 2:50 @rayradjr,
the usual thing is to ask you to post