Why we say loudness is proportional to the log of intensity,why log?

Intensity is a physical measure of acoustic power made with a meter.
Loudness is our perception of acoustic volume, ie how loud it sounds to us.

The smallest reasonably perceived change in loudness tends to be doubling or halving the power. In decibels, this is a 3dB change. 

This is hard to believe until you get on a mixing desk that is marked in decibels where the sliders are marked in 3dB increments and amazingly the difference in increments is only just noticeable - that is doubling the acoustic power with every increment! 

A perceived "twice as loud" is actually ten times the power

If you have a 1000W amplifier with a volume knob marked 0 to 10, for a smooth increase in perceived volume, the steps (in Watts) would be something like;
(0) 2 4 8 16 32 64 128 256 512 1024

That is where the logarithm comes in, perception versus physical reality.

Another way to broadly think about it is; if you are given the measured intensity or power, you don't care what the values of the digits are, you just look at how long the number is. Your ear deals with 10 versus 100 and 100 versus 10000, not 5 versus 8.

Try plotting the intensity of a whisper in a quiet room vs. the sound of a jackhammer on a linear scale.  The need for a logarithmic scale quickly becomes apparent when you compare the two.  A whisper is about 30dB, and a jackhammer is somewhere around 100dB.  For every 3 decibels, you get a doubling of power.
 That means that the sound of a jackhammer is over 8 million times more powerful than a whisper.  In this example, if we set the intensity of a whisper to where it was just visible on a linear graph, the intensity of the jackhammer would be a line that was over 8km tall.

Because appropriate tests and relevant measurements have shown that there is no direct proportionality between sound emitted intensity and what our ears perceive , that's why a logarithmic scale in decibels (dB).
Decibels are different from other familiar scales of measurement; while many standard measuring devices, such as rulers, are linear, the decibel scale is logarithmic.
This kind of scale better represents how changes in sound intensity are actually  felt by our ears.
To understand this, imagine a building 80 m tall. If we build up another 10 m, the building will be 12.5 percent taller, which would seem just slightly taller to us: this is a linear measurement.
Using the logarithmic dB scale (1 dB = 10*Log to base ten of the ratio I/Iref) , if to sound I of 80 dB  we add another 10 dB, the new sound I' will be 10^1 = 10 times more intense ( every 3 dB it doubles) 

We can define things however we like.  But mostly we are trying to make a scale that represents human experience.  Increasing a noise from 60 to 70 dB increases the energy by a factor of ten but it does NOT sound ten times louder.  Increasing it further to 80 dB only seems to add a similar increase as the one from 60 to 70. Which means that our human response is logarithmic.  And we discover that our response to light intensity is also logarithmic.  ie that every time we DOUBLE the light intensity it seems to our senses to increase by 1 unit.
There are so many things like this.  On my bike I have various cogs.  At the high end I have a ten tooth and an 11 tooth cog.  Changing between them gives a 10% alteration to pedalling speed and to force.
At the other end the two lowest cogs change from 30 to 34 teeth.  Even though the change is 4 teeth the RATIO is still a change of about 10%  So it "feels" to be about the same amount of change.
Someone earning $5per hour is given a raise to $7 per hour.  An increase of 40%.  But if you went to a CEO earning $5000 per hour and offered them the same $2 raise they would be insulted. Because the RATIO would be so much less.  Logarithmic scales are used any time that a similar change in ratio provides a similar "feel" to the person using the measurement.