[badlibrarian]

This essay needs a type system. Every time it says “log” it should say: log of what, into what?

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

[rq1]

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…

[eru]

> 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.

[jfengel]

I still don't understand why audio dB are negative. That's relative to what? What happens at 0dB?

[eru]

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]

See https://en.wikipedia.org/wiki/Apparent_magnitude

[deepspace]

0db is usually defined as the loudest sound that the audio system can produce. Hence, everything else must be negative.

[rdbl27]

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.

[mitthrowaway2]

Of course, typical of the wonderful ambiguity of decibels, 0 dB is also usually defined as the quietest sound that the human ear can perceive.

https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing

[ianburrell]

That's why important to give the scale. dBfs is full scale level, and db SPL is sound pressure level.

[kevin_thibedeau]

That is dB full scale where 0 is an absolute ceiling and you can deduct from there.

[jmyeet]

The first section details how the author thinks of "log N" with no base as an abstract object rather than a number. Or what are you referring to?

[badlibrarian]

The first section is the good part.

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?

[exmadscientist]

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.