I often find myself in need of a quick approximation to a base-2 logarithm of an integer, e.g. $\log_2 3$ or $\log_2 5$. While I can always reach for a calculator (or computer), I'd quite like to be able to derive one quickly using pen and paper, whenever I need it.
Ideally, such an approximation would be in the form of a sequence of rational numbers that approaches the target value, and which is easy to calculate by hand. Does anyone know of such a sequence?
To give an idea of what I mean: if I ever find myself needing to approximate the Golden ratio, I can simply write down the Fibonacci sequence, pick two consecutive terms and take the ratio between them. This gives successively better rational approximations to $\phi$ and I can easily calculate it even without a pen, since it requires only addition. While I imagine there isn't quite such an easy way to approximate $\log_2 3$ (or $\log_2 5$ or $\log_2 n$), I'm looking for something as close to that as possible.
(If such a thing exists only for natural logarithms it would still be helpful, but base 2 is greatly preferred.)