Does "order of magnitude" extend beyond base 10?

Question

Usually when talking about orders of magnitude, we're talking about base 10, but Benford's law, which only works if the numbers are spanning multiple orders of magnitude, works well in any base where the length of the smallest and largest numbers differ enough.

If we don't specify a base, it's hard to quantify what spans multiple magnitudes, since numbers in base 2 are three times longer than numbers in base 10.

Does the term "orders of magnitude" apply to other bases as well?
Is there a similar base-specific or base-agnostic term?

Of course, as long as you know what the base is that base is the magnitude. — Zelos Malum, Nov 15 '16 at 12:20
Taking a logarithm gives roughly the same value for a given base. — hardmath, Nov 15 '16 at 13:15

score 2 · Accepted Answer · edited Jun 05 '22 at 11:12

People do use "order of magnitude" to refer to base 2 orders, relying on context to make clear what units are being used. Yes, they are only 30% as big, but it's just a different unit for the same thing, analogous to the way feet are only 30% as big as meters.

In fact the analogy is exact when the orders of magnitude are themselves the units, as in the cases of bels (orders of magnitude are $10$), decibels (orders of magnitude are $\sqrt[10]{10}$) and nepers (orders of magnitude are $e\approx 2.71$).

In musical contexts it's common to measure sound frequency in orders of magnitude of size $2$ (called octaves), size $\sqrt[12]2$ (called semitones) and $\sqrt[1200]2$ (called cents).

Does "order of magnitude" extend beyond base 10?

1 Answers1