What is a logarithmic scale? Why is it more useful for some things than a regular numerical scale?
In: Mathematics
What you call a “regular” numerical scale works by addition. A log scale works the same, just with multiplication instead of addition. (To paraphrase mm..mBacon12345: 1, 2, 3, 4 becomes 1, 2, 4, 8.)
If you happen to be familiar with music, the musical scale is my favorite example of log scales in the “real world”. The keys on a piano are “evenly” spaced. Pick a note (say, middle C), and go one octave up (say, high C), and your second note’s frequency is double your first one’s. (That’s why they harmonize so well that we call them by the same name.)
There are 12 intervals in that octave (7 white keys, 5 black keys). If they were evenly spaced in an “additive” sense, a) it would sound awful, and b) the next octave up would sound awful, but not the same awful, a different awful. Instead, they are evenly spaced in a “multiplicative” sense, and a) it sounds great, and b) if you shift a tune one octave up, it is still recognizably the same tune.
To gloss over a long and interesting story, our idea of harmony is rooted in small ratios: 2/1, 3/2, 5/4, etc, and our modern 12-tone scale is kind of a best-fit of evenly-spaced small-number ratios between 1/1 and 2/1. The interval between two adjacent notes is the ratio 2^(1/12). So the piano keyboard is a log-scale number line.
Any topic where it’s the ratios of things that carry the meaning (rather than their sums and differences), will be at home on a log scale.
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