If the intensity is increased by a factor of 20, by how many decibels is the sound level increased?

If the intensity of sound in watts per square metre is I, then the intensity level in decibel (dB) is given by

dB= 10 log(I/I•),

Where the base of logarithm is 10,

I•=10^-12 watt per square metre( roughly the intensity at which audible sound can be heard)

On the decibel scale the pain threshold is 1 watt per square metre and on decibel scale is:

dB= 10 log(1 watt per square metre/10^-12 watt per square metre)

dB= 10 log ( 1/10^-12)= 10* log (10^12)=10*12=120 dB

In the given problem

dB= 10 log ( 20 I/I)= 10* log 20=10*1.3010=13dB