The logarithmic expression is the inverse of the exponential expression . The value is referred to as the “base”, which is chosen to be a positive real number.
The most common bases for logarithms are 10, , and 2. A logarithm with base is typically written as and is referred to as a "natural logarithm".
Logarithms have the following properties and identities:
(1)
(2)
(3)
(4)
(5)
(6)
A decibel (dB) is defined as one tenth of a bel (so there are 10 decibels in a bel). A bel is an amplitude unit defined as the of sound intensity relative to some reference intensity level. Since signal intensity, power, and energy are proportional to the square of signal amplitude, amplitude relationships in dB are given by of sound amplitude to a reference amplitude.
Logarithms can be useful for expressing relationships that occur over very large ranges on a linear scale in more manageable, intuitive terms. For example, the pressure “threshold of pain” of our auditory system is times greater than the "threshold of audibility". On a decibel scale, this range corresponds to 120 dB.
In signal processing, the maximum signal amplitude is typically chosen as the reference amplitude. In other words, signals are normalized so that the maximum amplitude is 1, or 0 dB.