So just what is a decibel (dB)? Probably not what you think.

Ask yourself this question: what exactly does a decibel measure? If you’re thinking “loudness of sound”, then you’re on the right track, but there’s more to it.

Any quantity can be measured on a decibel scale, but decibels are most useful when they describe an underlying quantity that varies over a vast range. Let’s take the example of “loudness”; this is formally measured by sound pressure. For example, the sound of a rifle being fired at a distance of 1m has a sound pressure of 200 Pa, whereas rustling leaves results in a sound pressure of 6 × 10-5 Pa (that’s 0.00006 Pa in decimal). We can hear both these sounds, though there is a factor difference of over a million in the measurement. However on the decibel scale, the rifle hits 140 dB and the leaves rustle at 10 dB, a much more manageable number range.

If you have some mathematical training, you’ll know that any large number range can be comfortably handled using logarithms. By definition, if 10a=b, then log b = a. Here are some examples:

\log\ 1000=3, \quad \log\ 10^9 =9, \quad \log\ 299 792 458 = 8.4768207.

To start with, you might begin by simply taking the logarithm of the underlying quantity. So let’s define “logbels” in this manner. For sound pressure as above, we’d have that

200\ \mathrm{Pa} = 2.30\  \mathrm{lB_{Pa}} \quad \text{ and } \quad 6 \times 10^{-5}\ \mathrm{Pa} = -4.22\  \mathrm{lB_{Pa}}.

Not a bad first attempt, but it suffers from two deficiencies. The first is that the logbel scale is dependent on the underlying units. For instance, if we’d chosen to measure sound pressure in pounds per square inch, we’d end up with different logbel units. The second problem is that the resulting numbers are difficult to interpret. Rustling leaves give minus 4.22 logbels.

A solution to both problems is to chose a reference measure and define decibels as the logarithm of the ratio of the underlying quantity with respect to the reference measure. Let’s revisit sound pressure. By convention, the underlying unit is Pascals squared, and the reference measure, X0, is chosen to be the auditory threshold at 2 kHz (that’s 2 × 10-5 Pa or 4 × 10-10 Pa2). Suppose we now measure the actual sound pressure and find it to be X Pa2. The decibel is then defined as

 X_{\mathrm{dB}} = 10 \log(X/X_0).

So with this new definition, the auditory threshold is 0 dB, placing the rustling leaves at 10 dB. An important property of logarithmic scales is that adding a constant to a number on the scale corresponds to multiplying the underlying quantity by a (different) constant. In the decibel scale, an increase of 3 dB corresponds to a doubling of the underlying quantity. The difference between leaves and a rifle is 130 dB, a factor difference of 2130/3. That’s about 243 ≈ (210)4 23 ≈ 8 × (103)4 = 8 × 1012. Remember, this is in squared units, so the ratio of the Pascal measurements is √(8 × 1012) ≈ 3 × 106, as before.

Wait a minute, didn’t I say that decibels weren’t just for sound? Well it should be clear from the formulation above that any quantity can be measured using a decibel scale, but decibels are most useful for measurements where human (or other) perception varies logarithmically with the underlying quantity. A common use for decibels is in measuring power in electrical circuits or of radio signals. Typically, the reference value will be a milliwatt, giving rise to the dBm (m for milliwatts). For example, if you have access to a wireless network, your wireless adapter may indicate received signal strength in dBm, here -50 dBm indicates an excellent signal, whereas -80 dBm is a poor signal.

There are other popular scales that are logarithmic in nature. Perhaps the best known is the “apparent magnitude” scale for measuring the brightness of stars. Our perception of brightness is logarithmic with respect to the received electromagnetic flux, leading to the following definition of apparent brightness:

m = -2.5 \log(F)+C,

where F is the flux and C is an appropriate constant. Note the negative sign in front of the logarithm. This means that smaller magnitudes correspond to brighter stars.

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