The core idea of music made with harmonic sounds is that “notes” with frequencies at small-integer ratios will “harmonize”. Harmonic sounds means something like a vibrating string where the vibrations are integer multiples of some fundamental frequency, because other non-integer-multiple vibrations are damped out by the fixture of the string at two points. Different (non-harmonic) types of sounds often sound better with a different sort of scale, for details see this book http://sethares.engr.wisc.edu/ttss.html
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The “octave”, 2:1, is the simplest whole-number ratio, and makes many of the vibrations in two notes in such frequency ratio align with each-other, to the point that two harmonic sounds exactly an octave apart almost sound like the same sound.
Other simple ratios like 3:1, 4:3, 5:4, etc. also “harmonize”, with (not quite as) many aligned overtones.
The core idea of the 12-note musical scale (pretty much regardless of specific tuning) is the approximation:
3^12 = 531441 ≈ 524288 = 2^19
3/2 ≈ 2^(7/12) [this is accurate to about 0.1%]
Or another way to say this: 7/12 of “doubling” on a log scale is very nearly “three-to-two”. Musicians call this ratio a “perfect fifth”.
In the case of equal temperament, an octave is split into 12 precisely equal steps (on a log scale), each one the 12th root of 2.
There’s one other nice approximation to take advantage of:
5^3 = 125 ≈ 128 = 2^7
5/4 ≈ 2^(4/12) [this approximation is only accurate to about 1%]
Musicians call this ratio a “major third”.
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Even outside music, these approximations can be useful for doing approximate computations.
If only our society switched from decimal to “duodecimal” numerals, it would be very natural to use logarithms base two, notated with “duodecimal” fractions.
If you have a number expressed in log base two, and you use duodecimal notation, approximately multiplying or dividing by 2, 3, 4, 5, 6, 8, 9, 10, 12, ... is very easy using addition/subtraction of easy-to-remember multiples of 2^(1/12).
Unfortunately our society instead has slide rules and measurement scales (decibels, etc.) which are all built around logarithms base ten, and decimal notation.