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.