Rational approximations of logs of small primes

Let’s say for some reason we need to approximate decimal logarithms of some small primes. We know 2^{10} \approx 10^3, so clearly log 2 \approx 3/10. (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

7^4 = 2401 ~ 2400 = 2^5 3^1 5^2.

Taking logs of both sides gives

4 L_7 \approx 5 L_2 + L_3 + 2 L_5

where L_x = \log_{10} x. And of course we have L_2 + L_5 = 1, so this relation becomes

4L_7 \approx 2 + 3 L_2 + L_3.

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 \log 224 \approx \log 225 and $\log 4374 \approx \log 4375$, namely

7 L_2 + L_7 \approx 2 L_3 + 2

and

5 L_2 + 7 L_3 \approx 4 + L_7

This is a system of three (approximate) equations in three unknowns, and solving it in the usual way gives

L_2 = 72/239 \approx 0.30125, L_3 = 114/239 \approx 0.47699, L_7 = 202/239 \approx 0.84519.

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 2^{10} \approx 10^3 (from familiar powers of two), or, taking 120th roots, 2^{1/12} \approx 10^{1/40}.
  • remember 2^{19} \approx 3^{12} (a fundamental fact of musical temperament).

That gives \log 2 \approx 3/10 right away, \log 3 \approx 19/12 \log 2 = 19/40, and then \log 2400 = 2 + 3 \log 2 + \log 3 = 3.375 so \log 7 \approx 3.375/4 = 0.84375. 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 2^{1/12}, the fact on which our usual musical tuning system is based: 2^{7/12} \approx 3/2, 2^{4/12} \approx 5/4, and a bit less accurately 2^{10/12} \approx 7/4. (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 2^{34/12} \approx 7, which gives \log 7 \approx 34/40 = 0.85. This is a less good approximation than those of \log 3 and \log 5, 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.

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