The decibel scale is a logarithmic scale for measuring the sound intensity level. Because the decibel scale is logarithmic, it changes by an additive constant when the intensity as measured in W/m2 changes by a multiplicative factor. The number of decibels increases by 10 for a factor of 10 increase in intensity. The general formula for the sound intensity level, in decibels, corresponding to intensity I is
β=10log(II0)dB,
where I0 is a reference intensity. For sound waves, I0 is taken to be 10−12W/m2. Note that log refers to the logarithm to the base 10.
Part A) What is the sound intensity level β, in decibels, of a sound wave whose intensity is 10 times the reference intensity (i.e., I=10I0)?
Part B) What is the sound intensity level β, in decibels, of a sound wave whose intensity is 100 times the reference intensity (i.e. I=100I0)?
Express the sound intensity numerically to the nearest integer.
β = dB
One often needs to compute the change in decibels corresponding to a change in the physical intensity measured in units of power per unit area. Take m to be the factor of increase of the physical intensity (i.e., I=mI0).
Part C) Calculate the change in decibels ( Δβ2, Δβ4, and Δβ8) corresponding to m=2, m=4, and m=8.
Give your answers, separated by commas, to the nearest integer--this will give an accuracy of 20%, which is good enough for sound.

The logarithmic scale has a great advantage when measuring magnitudes of a large number of scales, since it converts these values to linear, allowing easier viewing.

Part A

Let's look for decibels for an intensity I = 10 Io

We calculate

β = 10 log (10Io / Io)

β = 10 dB

Part b

Let's find the intensity for I = 100 Io

We calculate

β = 10 log (100Io / Io)

β = 10 log 100

β = 10 2

β = 20 db

Part c

Δβ2 corresponds to an intensity change of 10² Io, therefore it corresponds to an intensity increase of 10²

Δβ4 corresponds to a change in intensity of 10⁴Io

Δβ8 is an intensity change of 10⁸ Io