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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.

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Note that its easy to show that the function $g$ is a multiplicative map i.e; $g(xy)=g(x)g(y)$.. –  Deliasaghi 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. –  Deliasaghi Nov 11 '12 at 2:50
    
@rayradjr, the usual thing is to ask you to post your comment as an answer, with whatever detail you may wish. That way Ali can accept your answer. –  Will Jagy Nov 11 '12 at 3:00

1 Answer 1

By idea of @rayradjr, we can solve of my question, please see

Soon-Mo Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications 48, 2011, ISSN 1931-6828, page 227

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