How fast are the Taylor series of $\ln(1+x)$ and $\exp(x)$ converging to satisfy the $n$th decimal place? Is it dependend on the input $x$?

Question

How fast are the Taylor series of $\ln(1+x)$ and $\exp(x)$ converging to satisfy the $n$th decimal place? Is it dependend on the input $x$?

$x \in \mathbb R+$.

Maybe you can give an $O(n)$ for that - that means how many terms are needed $O(n)$ to satisfy the $n$th decimal place.

Are there any other good alternatives to calculate the $\ln(1+x)$, than just using tables and Taylor series?