This article is from the Audio Professional FAQ, by with numerous contributions by Gabe M. Wiener others.
Our ears respond logarithmically to increases in sound pressure
level. In order to simplify the calculations of these levels, as
well as the electrical equivalents of them in audio systems, the
industry uses a logarithmic system to denote the values.
Specifically, the decibel is used to denote logarithmic level above
a given reference. For instance, when measuring sound pressure
level, the basic reference against which we take measurements is the
threshold of hearing for the average individual, 10^-12 W/m^2. The
formula for dB SPL then becomes:
10 Log X / 10^-12 where X is the intensity in W/m^2
The first people who were concerned about transmitting audio over
wires were, of course, the telephone company. Thanks to Ma Bell we
have a bunch of other decibel measurements. We can use the decibel
to measure electrical power as well. In this case, the formula is
referenced to 1 milliwatt in the denominator, and the unit is dBm.
1 milliwatt was chosen as the canonical reference by Ma Bell. Since
P=V^2 / R, we can also express not only power gain in dB but also
voltage gain. In this case the equation changes a bit, since we
have the ^2 exponent. When we take the logarithm, the exponent
comes around into the coefficient, making our voltage formula 20
log. In the voltage scenario, the reference value becomes 0.775 V
(the voltage drop across 600 ohms that results in 1 mW of power).
The voltage measurement unit is dBv.
The Europeans, not having any need to abide by Ma Bell's choice for
a canonical value, chose 1V as their reference, and this is
reflected as dBV instead of dBv. To avoid confusion, the Europeans
write the American dBv as dBu. Confused yet? [Gabe]