The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor.
The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…
> The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation.
Units are important as a sort-of type system, even at the conceptual level.
You are right that bases are not as important conceptually.
The later reuse of “log” across valuations, dimension, vector fields, orders of vanishing is not so good. Those may be related ideas, but each needs a type signature: from what, to what, and preserving which operation?
Or, to say a little more explicitly what you're getting at: when you take a logarithm of some quantity, log x, x absolutely must be unitless. There's no way whatsoever to take a logarithm of something with a unit attached. (This is an important and useful dimensional analysis check in formulas and long calculations!)
So what do you do in practice? You have to normalize: you don't calculate log x, but instead log x/U for some scaling unit U. It's typical for U to be something like 1 mV or 1 W in electrical engineering, for example. This is completely legitimate, but it does mean that the thing that comes out needs a corresponding unit attached to it: dBmV, dBW, et cetera.
And it's really kind of important to be careful about that.
Well, the brightness of celestial objects is also sometimes negative:
> The apparent magnitude of known objects can range from −26.832 for our Sun to about +31.5 for objects in deep space imaged by the Hubble Space Telescope.[3]
And this is because Ptolemy’s catalog in which he ranked stars by their apparent brightness on a scale of one to six, one being the brightest. Ptolemy’s scale was (much later) retrofitted to a log scale (base 100^(1/5) or about 2.512), allowing extrapolation to both brighter and dimmer objects. The brightest of Ptolemy’s first-magnitude stars actually have negative magnitudes by the modern definition.
More specifically, 0 dB is the loudest sound the audio system is rated to produce without distortion. It's common to be able to actually drive systems harder than their specified engineering limits, which is why meters have a short positive dB section marked in red.
A sound source produces sound power and this generates a sound pressure fluctuation in the air. Sound power is the distance independent cause of this, whereas sound pressure is the distance-dependent effect.
Sound pressure p is a "sound field quantity" and sound intensity I is a "sound energy quantity". In teachings these terms are not often separated sharply enough and sometimes are even set equal. But I ~ p2.
It’s like audio where people say "dB" as if it answers the next question. Relative to what, measured how, and weighted for whom?
Author should brush up on https://en.wikipedia.org/wiki/Lie_theory