Functional equation with logarithm

Question

I have $\dfrac{g(x)}{g(y)}=\dfrac{\log x}{\log y}$. Can I conclude from this that $g(x) = k\log x$ ? I have very little knowledge of functional equations.

Any help appreciated.

Answer 1

You have $$\frac{g(x)}{\log x}=\frac{g(y)}{\log y}.$$ The LHS is independent of $y$ and the RHS of $x$. Thus both sides are surely constant?

Answer 2

For $x >0$ and $x \ne 1 $ we have:

$\dfrac{g(x)}{\log x}=\dfrac{g(e)}{\log e}=g(e)$.