Suppose that $L:(0,\infty)\rightarrow \mathbb{R}$ is differentiable, that $L(1)=0$, and that $L'(x)=1/x$ for all $x>0$.
Then $L(2^n)>n/2$ for all $n\in N$.
This implies that $\sup(L)=\infty$, and a similar argument shows that $\inf L = -\infty$. By the inverse-function teorem, it follows that $L$ is a bijection from $(0,\infty)$ onto $\mathbb{R}$ and that its inverse $E=L^{-1}$ is differentiable. Show that $E'(t)=E(t)$ for all $t\in\mathbb{R}$.
Obviously the function is the natural log, but for this I can not use the properties of $\ln$ nor integration. So how do I show this without knowing what the function is or its inverse?