Join the discussion
Write your take first — we'll ask for email only when you're ready to publish.
- Hacker News
- The same idea comes up in physics. In quantum physics, the action S appears as the logarithm-like quantity behind the amplitude e^iS/(h^bar). In statistical mechanics, entropy is the logarithm of the number of possible microstates Omega : S = log(Omega). Although the concepts come from different parts of physics, they both reflect the same principle: using a log as a way to turn multiplicative relationships into additive ones.by GL26
- >You might ask: if we have a baseless logarithm log(N), do we also have a “baseless exponential”?
Sure we can, with some naive algebra. If we can take log(x,base) and drop the base, then we can also take pow(base,x) and drop the base. Since bits=log(2), then pow(bits)=2. You can probably connect it to the reverse of things, like integrals.
Also, for fun, I'll play with some notation tricks.
I like the intuition that baseless logarithm notation gives, and it also avoids needing to choose a specific reference point. I can also directly calculate by choosing an arbitrary base:log(freq) = pitch freq = pow(pitch) octave = log(2) 400*Hz = 100*Hz*4 // the frequency 400 Hz equals 4 times 100 Hz log(400*Hz) = log(100*Hz) + log(4) log(400*Hz) = log(100*Hz) + 2*log(2) log(400*Hz) = log(100*Hz) + 2*octave log(400*Hz) = log(100*Hz) + 2*octave // the pitch of 400 Hz equals 2 octaves above the pitch of 100 Hz cent = log(2)/1200 A4 = log(440*Hz) B4 = A4 + 200*cent // the pitch B4 equals 200 cents above A4 B4 = log(440*Hz) + 200*log(2)/1200 B4 = log(440*Hz) + log(2^(2/12)) B4 = log(440*Hz * 2^(2/12)) pow(B4) = 493.883 Hz // the frequency of B4 equals 493.883 Hzpow(log(440*Hz) + 200*log(2)/1200) exp(ln(440) + 200*ln(2)/1200)by amavect - Charles Petzold's The Lost Art of Logarithms is a great read (still a work in progress).by orc00
- I think what's going on with the complex logarithm is basically the same as the logarithm that outputs the set of all possible bases for a vector space. The complex logarithm produces a Z-torsor, and the basis logarithm produces a GL(V)-torsor. There's probably some way to represent a choice of branch cut as a part of the choice of the base of the complex logarithm, and similarly the choice of a specific basis as part of the choice of base of the vector space base logarithm.by aesthesia
- 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
by badlibrarian - That's a lot of ways to think about logarithms.
Logarithms are laughably simple once you've fully internalized the meaning of the log function; it simply answers the question:
"To what power must I raise the base to get the argument?"
This is why the output tapers out as you increase the argument; because even if you increase the argument exponentially, you only need a fixed increment in the power to reach that number... So if you increase the argument only by a fixed amount (linearly) instead of exponentially, then it makes sense that the output will grow sub-linearly.
I remember when I was doing algebra with logs many years ago at school, I was applying rules to remove the log from one side of the equation.
Then when I got to uni, I had to revise the rules but it was kind of silly of me because those rules can be trivially derived if you just think about what the log function means. Turns out I had been solving equations with logs throughout school without understanding what they even meant... It's only at university that I actually bothered to learn them.
Actually TBH. I didn't even fully understand powers for some time even though I was doing calculus with them at school. I only fully understood powers once I properly internalized the concept of k-ary trees as a proxy.
It's one thing to be able to apply something, another to understand it. And I think to innovate with something, as a tool, it's not enough to be able to apply it. You must understand it.
by jongjong - Logs are awesome. I started a math textbook from the 1920's a while ago, and all the calculations relied on tabulated logs, where you would convert the number to a log in a table to reduce the operation's degree, then convert back to the ordinary representation. This would reduce operations like finding cubed roots to division, would could be converted to log-log to be further reduced to subtraction before you would restore to ordinary notation. It feels like you're using a magic wormhole or something when you're doing this stuff by hand, it's really neat.
- The baseless log here is just a torsor [0]!
Lots of things are torsors: position, currency values, calendar dates etc. the vales themselves are arbitrary, and translating/scaling them by some value doesn't make a functional difference. Torsors let us talk about these things without needing to make such an arbitrary choice a priori.
In the case of baseless logs, the underlying set is "information units", i.e. log 2 is bits, log e is nats, log 10 is digits, etc. The conversion factors give us the torsor's group, and picking a privileged unit is just a trivialization of the torsor.
The vector division notation is, similarly, encoding a g-torsor in precisely the same way as length units are.
The examples so far are all torsors with abelian groups, but specifying position both requires choosing an origin and a length unit. The group of this torsor is a suitable semidirect product between translation and scaling, which gives a non-abelian group.
Most of the time we just implicitly choose a trivialization, which often causes confusion because it identifies objects with operations on them, e.g. conflating vectors as positions with vectors as translations. The author's treatise on problems with geometric algebra [1] even brings up this point!
[0]:https://math.ucr.edu/home/baez/torsors.html
[1]:https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...
by xelxebar