Derive the relation between Intensity and loudness.

Explanation

According to Weber- Fechner's law, "loudness (L) is proportional to the logarithim of the actual intensity (l) measured with an accurate non-human instrument". This means that
`L propto ln I`
L=k ln I
where k is a constant, which depends on the unit of measurnment. The difference between two loudness, `L_(1)` and `L_(0)` measures the relatie loudness between two precisely measured intensities and is called as sound intensity level. Mathematically, sound intensity level is
`Delta L=L_(1)-L_(0) = k ln I_(1) -k ln l_(0) = k ln [l_(1)/l_(0)]`
If k=1, then sound intensity level is measured in bel, Therrefore,
`Delta L = ln[I_(1)/I_(0)]` bel.
However, this to express smaller unit, decibel, Thus, [decibel `=1/10` bel] by multiplying and dividing by 10,
`Delta L = 10(ln [I_(1)/I_(0)]) 1/10` bel
`Delta L=10 ln [I_(1)/I_(0)]` decibel with k=10
For practical purposes,
`Delta L = 10 log_(10) [l_(1)/l_(0)]` decibel.