Series approximations to e to the x and ln x

How would you approximate $e^x$ and $\ln x$ numerically, and what can be said about the size of the error?

Truncated Taylor series. $e^x\approx\sum_{k=0}^{n}x^k/k!$ with Lagrange remainder $e^{\xi}|x|^{n+1}/(n+1)!$; $\ln(1+u)\approx u-u^2/2+\dots\pm u^n/n$ for $|u|<1$, where for $0<u\le1$ the alternating-series error is below the first omitted term $u^{n+1}/(n+1)$. In practice one first reduces the argument (e.g. $e^x=(e^{x/2^m})^{2^m}$, $\ln x=m\ln2+\ln y$) so the series converges fast.