Suppose that we have the functional equations ("Abel equation", it is called) for a function $F: [1, \infty) \rightarrow \mathbb{R}$ given by
$$F(1) = 0$$ $$F(ex) = F(x) + 1$$
where $e$ is the natural logarithmic constant. Suppose $F$ is twice-differentiable, i.e. $C^2$. What function $F$ satisfies these conditions and minimizes the "energy"
$$E[F] = \int_{1}^{\infty} [F''(x)]^2 dx$$
? Is there such a minimizing function? If so, what is the proof? Does $F(x) = \log(x)$ work? If not, what does? If not, then what functional $\mathfrak{F}$ does $F(x) = \log(x)$ minimize for these equations?
A little work shows that f is C^infinity. If we assume it is analytic then using existence and uniqueness theorem of first order differentable equations and analytic continuaion from complex analysis and since we have at hand log z satisfies the same conditions we see that log z is the only solution.