It is sometimes handy to be able to estimate the

natural logarithm (

ln) of a number. This is actually not too hard, if you adapt a

trick from

Knuth: the

base-2 logarithm is almost exactly the sum of the natural and

common logarithms.

log_2(x) ~= log_10(x) + ln(x)

It's easy to check this formula, and see that it gives ~1% accuracy.

So let's say you're thinking about the density of prime numbers of size ~10^{600}. This is essentially asking what's `ln(1e600)`, so let's use our estimate. `log_10(1e600) = 600`, obviously. And we also know that `2^10 = 1024 ~= 10^3`, so `log_2(1e600) ~= 2000`. So `ln(1e600) ~= 1400` (the correct answer is `~1381.55`, so even after fudging the base-2 logarithm, we didn't do too badly!