Strange Logarithmic Coincidences

I was planning to write a post about the Rule of 72 for computing the doubling time of an investment, but got sidetracked when I saw this post from John D. Cook‘s blog about logarithmic coincidences. Apparently, Donald Knuth, in The Art of Computer Programming, notes that the three most commonly-used logarithms share the following relationship:


So, the base-2 logarithm can be approximated by the base-e and base-10 logarithms. Here are the corresponding graphs:


The difference barely registers on this scale. Using a simple change-of-base formula, we see that


which means that the real source of the approximate values can be expressed with the natural logarithm only:


The relative error here is about 0.00582. Naturally (pun intended), I was curious: are there other trios of bases that give even better approximations? (Though since this is a genuine mathematical coincidence, maybe it’s better to call them near-equalities.) After messing around in Sage for a while, I found that the answer is clearly yes. Here are a few more logarithmic coincidences for you…


The first one isn’t really a coincidence at all, since (4, 8, 64) = (22, 23, 26) means that the equation reduces to a decomposition of unit fractions: 1/2 = 1/3 + 1/6. So that’s a for-real equality, not just a near-equality. The others appear genuine, though. I’ve discovered dozens of these so far, but the third one is the most accurate near-equality that I’ve found.

I’m not sure what all of this means. Maybe these logarithms closely approximate some certain fraction decompositions, but maybe they are just a collection of coincidences drawn from the set {ln(n) | n > 1} of transcendental numbers. Either way, I’ll keep you updated on any new discoveries.


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