Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? 

For example, they can define $e=\lim\_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition how do they prove:

* It converges!
* $\frac{d}{dx}e^x=e^x$!
* etc.!

Then if we know $e^x$ injective from $\mathbb{R} \rightarrow\mathbb{R^+}$, we can call $\ln(x)$ the inverse of it. If we can prove $\ln$ is differentiable on its domain, then we can say: $$1=\frac{d}{dx}x=\frac{d}{dx}e^{\ln(x)}=e^{\ln(x)}\cdot\ln(x)'=x\cdot\ln(x)\Rightarrow\ln(x)'=x^{-1}$$ but this all depends on the above.