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
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…
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?
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.
"Sound Power Level SWL", "Sound Pressure level SPL", and "Sound Intensity Level SIL" are different quantities which should not be confused. - https://sengpielaudio.com/calculator-soundpower.htm
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.
> 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]
Understanding dB - http://www.jimprice.com/prosound/db.htm
dBFS - https://en.wikipedia.org/wiki/DBFS
Videos:
Understanding dB level by Paul McGowan - https://www.youtube.com/watch?v=t3Via4c8SlI
Paul explains 0dB and why there's a minus sign on volume - https://www.youtube.com/watch?v=NgEr6NEDPd4