The relationship between intensity levels and sound intensities is expressed using logarithmic scales. Specifically, intensity level (IL) is measured in decibels (dB) and is given by the formula:
IL = 10 * log10(I / I0)
where I is the sound intensity being measured and I0 is the reference intensity, which is typically the faintest audible sound.
If the intensity of the louder sound is double the intensity of the faintest audible sound, we can express it as:
I_louder = 2 * I_faintest
Now, let's calculate the increase in intensity level:
IL_louder = 10 * log10(I_louder / I0) = 10 * log10((2 * I_faintest) / I0) = 10 * log10(2) + 10 * log10(I_faintest / I0) ≈ 10 * 0.301 + IL_faintest
Using the property log10(2) ≈ 0.301, we can simplify the equation.
So, the intensity level of the louder sound would increase by approximately 3.01 dB compared to the intensity level of the faintest audible sound.