Give me a clear explanation on the relation of log with sound waves.?

Human senses are logarithmic, other wise we would be deaf and blind to week sound and lights and easily overloaded by strong sound and lights. Sound is measured in Bels, where a Bel means 10 times stronger, ie the log of which is +1 larger. To define a more precise measurement, we use deci-Bels. Bels or decibels is a measure of power. Power is the product of force/voltage and current/speed, where the current/speed varies as the voltage/force so power varies as the square of force/voltage. This is important because often the force or voltage of a wave is what is referred to as the intensity, instead of talking about power. That is why you may see a formula for dB that uses 20log instead of 10log.

For absolute measurements, the reference sound level of zero dB is based on the threshold of human hearing, sounds over 100dB can damage your hearing and waves that create a total vacuum are around 194dB, which would kill you.

The range of frequency which the human ear is designed to hear is 20Hz to 20Khz. This is denoted as sound.

The sound comming from a sound source has a certain amplitude. In audio and music systems we can control the amplitude (level) of the sound. However, there are other natural and man made sounds which cannot be controlled (eg. thunder, blast, crackers etc.)

The human ear has a built in protective system to accomodate the change in amplitude. Its response is lograthimic, not linear. For example, if a sound has an amplitude of x, and if the amplitude is increased by 2x, the ear will not hear it twice as loud. If that was the case, an amplitude of 10x would blast away our eardrums.

Hence the lograthimic scale of decibels (db) is used for calculations involving sound.

Hope your question is answered.