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 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]
"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.
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
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.