When a sound gets louder the increase in
loudness is proportional to the fractional change in intensity or:

Loudness increase
(dL) is proportional to increase in intensity (dI)/original intensity (I)

This means that
equal increases in loudness are the result of equal proportional changes in intensity. So if the
intensity of a sound doubles and then doubles again the loudness will increase by the same
amount each time.

Writing this mathematically: dL is proportional to dI/I or dL =
kdI/I where k is a constant.

If we integrate this we get the equation: L = k
ln(I/I_{o}) or L = C log(I/I_{o}) where I_{o} is the threshold
intensity, or initial intensity, and k and C are constants.

This shows the logarithmic
response of the ear to the volume of a sound.

If the intensity of a sound
goes up from say 2x10^{-8} Wm^{-2} to 4x10^{-8} Wm^{-2} and then
increases again from 4x10^{-8} Wm^{-2} to 8x10^{-8} Wm^{-2} the
loudness will increase by a factor of 2 each time.