A lot of the time, it's more useful to work with ratios, rather than absolute numbers. For instance, an amplifier has a gain of 100; that is the output is 100 times as big as the input. Or, for propagation through some (lossy) medium, the signal strength decreases by some ratio for passage through a certain distance. Double the distance, and the ratio is squared. When ratios come in, using a log scale is a natural choice, because multiplying and dividing by ratios is like adding and subtracting their logs.
The big question is then, which base for the logs. The by far and away most popular base is 10, which leads to a unit called the Bel. (Yes, named for Alexander Graham Bell, inventor of the telephone.). And, because folks like to work with integers, and jumps of a factor of 10 are pretty big, a more popular variant called the decibel (dB). Watch out for the form depending on whether you are working with powers (usual), or with voltages or currents and an assumed constant impedance (where power varies as the square of either).
dB = 10 * log10 (power ratio)
power ratio = 10^(dB/10)
db = 20 * log10(voltage ratio) -or- dB = 20 * log10(current ratio)
voltage or current ratio = 10^(dB/20)
The other likely candidate for the base of the log is e (=2.71828...), which has nice properties in equations, etc., and just seems to fall out of all sorts of natural phenomena (unlike 10, which is popular because we have 10 fingers). Using e as your base leads to a less common unit called the neper (Np). Neper is a form of Napier, the guy who popularized the use of logs for computation (viz. Napier's Bones). Nepers are used with voltage (or magnitude) ratios, so watch out when converting back and forth between Nepers and dB.
Np = ln (voltage ratio)
voltage ratio = exp(Np)
You can convert back and forth pretty easily using standard log and exponent identities: log[base y](x) = log(x)/log(y).
dB = 20 * log10( exp(Np)) = 20 * ln(exp(Np)/ln(10) = 20/ln(10) * Np = 8.688...
* Np
Np = ln(10^(dB/20)) = ln(10)*exp(ln(dB/20)) = ln(10) * dB/20 = ln(10)/20
* dB = 0.115129.. * dB
Here are some really long values for the constants (handy for cutting and pasting):
Np = dB * 0.115129255
dB = Np * 8.685889638
http://www.its.bldrdoc.gov/fs-1037/dir-024/_3502.htm definition of Neper
http://www.its.bldrdoc.gov/fs-1037/dir-010/_1468.htm definition of dB
http://www.its.bldrdoc.gov/fs-1037/dir-004/_0577.htm definition of bel
radio/neper.htm - Revised 5 November 2001, Jim Lux
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