Find all entire functions $f$ of finite order such that $f(log(n)) = n$
I am reviewing for my complex analysis final and this problem came up towards the end of Conway and I was unsure how to solve it. Can someone show me how to do this and potentially also the process for other questions of this type?
Any help is appreciated.
Sketch of the result that $f(z)=e^z$.
Since $e^{\log n}=n$, one has $g(\log n)=0$ for the finite order function $g(z)=f(z)-e^z$; but if $g$ (non identically zero) has order $a$ then in the disc of radius $R$, it has $O_{\epsilon}(R^{a+\epsilon})$ zeroes (an easy application of Jensen theorem and standard stuff in the theory of entire functions) and that of course contradicts the fact that $\log n$ are zeroes as those grow exponentially in number with $R$ (as $\log n \le R$ means $n \le e^R$)
hence $g=0$ and $f(z)=e^z$