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