10dB: a factor of 10 in Watt? Power or Amplitude?

Is it 10 or 20 dB that corresponds to a factor of 10 ... ?

In power or in amplitude ... ?

These are common head-scratchers that a physicist might have when faced with dBs (they come up every 5 years or so).  Okay, let's figure this out so that in 5 years I can come back to this place and read up on it.

Let's start with what we know.  Calculating dBs goes like this:

dB = # log(ratio)

The log is in base 10.  The ratio is either one of amplitudes or powers.  And there is a pre-factor # that is either 10 or 20, but we can never remember.

To figure this out, let's realize that the number of dBs (say 10 dB) must always describe the same thing about a signal, whether we were measuring in power or amplitude.  A signal that changes 10 times in power will change sqrt(10) times in amplitude.  So, what are the prefactors?

dB(power) = dB(amplitude)

#power log(10) = #amplitude log(sqrt(10))

#power = #amplitude * 1/2

So #power must be the one equal to 10, and #amplitude must be the one equal to 20.

dB = 10 log(power ratio)
dB = 20 log(amplitude ratio)

i.e.

20 dB is a factor of 10 in amplitude
10 dB is a factor of 10 in power

and

6 dB is a factor of ~2 in amplitude
3 dB is a factor of ~2 in power

whew.