[amavect]

>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.

  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 Hz

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:

  pow(log(440*Hz) + 200*log(2)/1200)
  exp(ln(440) + 200*ln(2)/1200)

[amavect]

Hah, I can use this to give decibels an actual unit.

  dB_P = log(10)/10
  dB_F = log(10)/20
  log(10*V) = log(V) + 20*dB_F  // the level of 10 V equals 20 dB more than the power level of 1 V.

  SPL = 20*10^-6 * Pa
  hearing_damage = log(SPL) + 90*dB_F  // hearing damage occurs over 90 dB_F above SPL (neglecting A-weighting)
  pow(hearing_damage) = pow(log(SPL) + 90*dB_F))
  pow(hearing_damage) = pow(log(SPL) + 90*log(10)/20))
  pow(hearing_damage) = SPL*pow(90*log(10)/20))
  pow(hearing_damage) = SPL*31622.7766  // the pressure of hearing damage occurs above 31622 times SPL
  pow(hearing_damage) = 0.632455532 Pa  // the pressure of hearing damage occurs above 0.632 Pa

Very helpful!! Imagine combining the goofy list of decibel suffixes into a uniform notation. Write the logarithm first so the + or - stays in the same spot.

  log(reference_unit) + value*dB_F (or dB_P)
  log(reference_unit) - value*dB_F (or dB_P)

https://en.wikipedia.org/wiki/Decibel#List_of_suffixes

[Rotundo]

Amazing. Thank you for giving me a new mental tool.

[ajkjk]

True, I guess you can just 'curry' exponentiation and say that's a baseless power. I couldn't find a clean notation for it so I gave up..