Logarithm

From Inspired Acoustics Knowledge Base

The logarithm was introduced by John Napier in 1614. It is a mathematical operation that transforms a multiplication into an addition and the power operation into multiplication. It is used in music, audio and acoustics as a nonlinear scale corresponding to sound perception, hearing and representation in fractional octave bands. It is also used in Decibel scale.

Explanation

In its most simple sense, a logarithm is an exponent. For example when you annotate the number 100 as being 10 "squared", the numeral 2 in superscript is the exponent, and the actual value of the logarithm (or exponent) is 2 in this example. Logarithms are important because they make it easy for one to multiply or divide large numbers. For example, if you wished to multiply, say 3 million by say 200,000, in real life, you would simply multiply the 3 x 2 and get 6, and then add up the numbers of zeros. In this example, three million has six zeros and 200,000 has five zeros. So your final answer would be 6 with 11 zeros after its or 600 billion (600,000,000,000).

Now the above mathematical example was ridiculously simple, because you didn't need logarithms to do the multiplication. However, the notation of logarithms would have been as follows: the logarithm of 200,000 is 2 x 10 to the fifth power (the logarithm is 5 in this simple example); similarly three million is 3 x 10 to the sixth power (the logarithm is 6 in this case). Therefore, you simply multiply 2 times 3 to get 6, and then you ADD the logarithms (add the exponents is the same as multiplying) so the final logarithm is eleven.

When we use the decibel scale for sound pressure level and its need for logarithms, we must first discuss a little background. Regardless of how loud or soft a given sound is, when produced by an audio amplifier, if you DOUBLE the amplifier power, you get an increase in 3 decibels (dB). This difference is hardly noticeable in real life. For example, in a stereo system, if you were to listen only to the left channel of a mono signal source at, say one watt of power, the sound would have a certain loudness when you are a certain distance away from the single loudspeaker.

Now, if you were to turn on the right channel, also with one watt of amplifier, you would be clearly hearing the audio result of two watts of power (one from each channel). However, when you add the second channel's worth of sound to the mix, you will be amazed at how little louder the doubled signal is.

Generally speaking, humans perceive a so-called "doubling of loudness" at about a 10 dB increase in sound pressure level. At the same time, the amplifier requires approximately TEN TIMES the amplifier power to incur a 10dB signal increase. This is massive! What this means is that if you have a certain loudness of sound coming from a one-watt signal, it will "sound" twice as loud when the amp is putting out ten watts of power. But it will only sound twice again as loud when the same amp puts out 100 watts! To sound twice as loud again, the amp needs to pump out 1000 watts!

Here is where the logarithms come in. When people talk about a 60dB change in level for reverb to die away, we are talking about a logarithm of 6 (10db times 6 to achieve a 60dB change). Guess what a difference of 60dB is? Yes, that's right, it is a change of one million times in amplifier signal. In terms of reverb time to achieve -60 dB, we are talking about the time it takes for the original audio signal to decay by one million times. More about the reverberation time here.