It is sometimes handy to be able to estimate the natural logarithm
) of a number. This is actually not too hard, if you adapt a trick
: the base-2 logarithm
is almost exactly the sum of the natural and common logarithm
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 ~10600. 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!