The bel is the logarithm to the base 10 of one measurement of a physical quantity relative a reference level. Decibel is one tenth of a bel.
An increase of 1 decibel in the measurement of a physical quantity implies the quantity increasing by 10 times.
Let the reference for measuring the intensity of noise be N.
At 10:00, let the noise intensity in the cafeteria be N_10. This in terms of decibel is 50 dB
=> `10*log_10(N_10/N) = 50`
=> `(N_10/N) = 10^5`
=> `N_10 = 10^5*N`
At 12:00, the noise in terms of decibel is 100. Let the intensity of the sound be N_12
=> `10*log_10(N_12/N) = 100`
=> `(N_12/N) = 10^10`
=> `N_12 = 10^10*N`
`N_12/N_10 = 10^10/10^5`
The intensity of sound increased by a factor of `10^5` .
Let `I_1` be the intensity at 10:00,`I_2` the intensity at 12:00.
` `Note: The level of sound, measured in decibels (dB), is found by comparing the intensity of a sound (measured in watts per square meter) to the threshold of human hearing `I_0=10^(-12)W/m^2` .
Thus `L=10log_(10)(I/I_0)` .
(3) So the sound intensity increases by a factor of `I_2/I_1=10^22/10^17=10^5`