The decibel is a scale which is used in acoustics to measure sound levels. The zero of the scale is the threshold of hearing which is approximately the intensity of sound created by a mosquito flying 3 m away. This is taken as the lowest sound level that an average person can hear and given a value of 0 dB.
The decibel scale is a logarithmic one where an increase of 10 decibels increases sound intensity ten times. For example, a sound level of 20 dB has an intensity 10 times that of a sound level of 10 dB .
In the given problem Aaron plays music at 80 dB while the apartment building has an upper limit of 40 dB. The intensity of the sound from Aaron's stereo is 10^[(80 - 40)/10] = 10^4 = 10,000 times the acceptable limit.