Let’s say for some reason we need to approximate decimal logarithms of some small primes. We know , so clearly . (All logarithms in this post are to base 10.) Can we approximate the logs of any other primes this way?
When I run through powers of small primes, the target that jumps out at me is
Taking logs of both sides gives
where . And of course we have , so this relation becomes
Are there other such relations? It turns out that there are finitely many numbers n such that n and n+1 have all their prime factors < 7 – apparently a conseuqence of the abc conjecture – so I just take the largest ones and derive relations like these. So I can derive similar relations from and $\log 4374 \approx \log 4375$, namely
This is a system of three (approximate) equations in three unknowns, and solving it in the usual way gives
while the true values are 0.30103, 0.47712, and 0.84510 – so this is good enough for three-place accuracy.
In practice, who wants to have a rational with 239 in the denominator as an approximation? More practical is the following:
- remember (from familiar powers of two), or, taking 120th roots, .
- remember (a fundamental fact of musical temperament).
That gives right away, , and then so . This is essentially I. J. Good’s singing logarithms, which exploits the fact that a lot of rational numbers with small numerator and denominator can be approximated as powers of , the fact on which our usual musical tuning system is based: , and a bit less accurately . (Musically these are the facts that the perfect fifth, major third, and minor seventh are the ratios 3/2, 5/4, and 7/4.).
From the last we can derive , which gives . This is a less good approximation than those of and , just as the seventh harmonic isn’t really a note in the equal-tempered scale – the harmonic or “barbershop” seventh is noticeably flat compared to the minor seventh.