Logarithm
Definition
The inverse of exponentiation — it answers ‘to what power?’ and tames quantities spanning many orders of magnitude.
Technical definition
log_b(x) = y means b^y = x. Natural log (ln, base e) appears in decay, growth, and entropy; base-10 logs define the decibel and Richter scales. Key rules: log(ab) = log a + log b; log(a^n) = n·log a.
Example
A 20 dB sound is not ‘twice’ a 10 dB sound — decibels are logarithmic, so 20 dB carries 100× the intensity of 10 dB.
Common confusion
You cannot take logs of units directly — always form a dimensionless ratio first (like I/I₀ in decibels). log(5 meters) is meaningless; log(5 m / 1 m) is fine.