TITLE: Calculating logs and fractional exponents by hand
QUESTION [20 upvotes]: In view of what we can compute by hand, on a piece of paper, without having to use a computer or a calculator, how far can we go with the evaluation of $\log$-functions and fractional powers?
More concretely, are there practical methods, that work in general well enough, for computing the followings?

$\log(x)$ (natural log used here) for $x\in ]0,\infty[$ (e.g take $x=3,$ which has a natural log of $\approx 1.09$)
$x^{\alpha},$ ($x$ being any real number) for $\alpha$ not being a natural number. So e.g. $\alpha=1/2,1/3,\cdots$

It would be neat to learn about possible ways of maybe first simplifying these calculations, translating them into more feasible calculations (by hand), and then take it from there. Or if direct methods exist that work for certain values or powers. This surely begs the question, how people used to do these calculations before computers were around.

REPLY [10 votes]: My favorite way of calculating the natural logarithm:
$$\ln(x)\approx-\gamma+\sum_{n=1}^x\frac1n$$
Where $\gamma$ is the Euler-Mascheroni constant.  Not only is this easy, but it works quite well for something like $x>10$.
If you need to calculate other bases, like $\log_b(x)$, use $\log_b(x)=\frac{\ln(x)}{\ln(b)}$.
If you are not satisfied with the result because $x$ is small, try $\ln(x)=\frac1k\ln(x^k)$ to make the approximation better.