Question 1038963: The loudness level of a sound can be expressed by the comparing the sound's intensity to the intensity of a sound barely audible to the human ear. The formula D=10(log I-log I). Describe the loudness level of a sound, D, in decibels, where I is the intensity of the sound, in watts per meter^2, and I is the intensity of a sound barely audible to the human ear. A) express the formula so that the expression in parentheses is written as a single logarithm. B) use the form of the formula from part (a) to answer this question. If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you need a different variable to represent the intensity of the larger sound than the variable that represents the intensity of the smaller sound.
capital I for the larger sound and small i for the smaller sound would do it.
the formula is D = 10 * (log(I) - log(i))
since log(a) - log(b) = log(a/b), then D = 10 * (log(I) - log(i)) is equivalent to D = 10 * log(I/i).
the formula becomes D = 10 * log(I/i).
if the intensity of a sound is 100 times the intensity of a softer sound, then I can be replaced with 100 * i and the formula D = 10 * log(I/i) becomes D = 10 * log(100*i/i) which becomes D = 10 * log(100).
log(100) is equal to 2 because 10^2 = 100.
the formula becomes D = 10 * 2 which makes D = 20.
a sound 20 DB higher than another sound is 100 times as intense as that of the other sound.
the following reference supports this conclusion.
http://www.allaboutcircuits.com/textbook/semiconductors/chpt-1/absolute-db-scales/
the table in this article has 1 milliwatt of power as a reference.
you'll need to scroll down a little until you find it.
1 milliwatt is not shown, but it is assumed to be 0 db.
0 db is equal to 1 milliwatt which is equal to .001 watts.
100 milliwatts is shown as 20 db.
100 milliwatts is 100 times the intensity of 1 milliwatt of power.
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