SOLUTION: The loudness, b, of sound measured in decibals is defined by the equation b = 10 log [I/Io] where "I" is the intensity of the sound and "Io" is the minimum intensity detectab

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: The loudness, b, of sound measured in decibals is defined by the equation b = 10 log [I/Io] where "I" is the intensity of the sound and "Io" is the minimum intensity detectab      Log On


   

Question 634678: The loudness, b, of sound measured in decibals is defined by the equation
b = 10 log [I/Io]
where "I" is the intensity of the sound and "Io" is the minimum intensity detectable. Show that, if the difference in loudness of two sounds is "d" decibels, the louder sound is 10^d/10 times more intense then quieter sound.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let I%5B1%5D be the intensity of the louder sound and I%5B2%5D be the intensity of the quieter sound. This makes the loudness of the louder sound:
10%2Alog%28%28I%5B1%5D%2FI%5B0%5D%29%29
and the loudness of the quieter sound would be:
10%2Alog%28%28I%5B2%5D%2FI%5B0%5D%29%29

Since "d" is the difference of loudness of these sounds:
d+=+10%2Alog%28%28I%5B1%5D%2FI%5B0%5D%29%29+-+10%2Alog%28%28I%5B2%5D%2FI%5B0%5D%29%29
We are now going to solve this equation for I%5B1%5D (so that it will make a statement about the intensity of the louder sound. Factoring out 10 we get:
d+=+10%2A%28log%28%28I%5B1%5D%2FI%5B0%5D%29%29+-+log%28%28I%5B2%5D%2FI%5B0%5D%29%29%29
Dividing both sides by 10:
d%2F10+=+log%28%28I%5B1%5D%2FI%5B0%5D%29%29+-+log%28%28I%5B2%5D%2FI%5B0%5D%29%29
Next we can use a property of logarithms, log%28a%2C+%28p%29%29=log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29, to combine these logs:
d%2F10+=+log%28%28%28I%5B1%5D%2FI%5B0%5D%29%2F%28I%5B2%5D%2FI%5B0%5D%29%29%29
The I%5B0%5D's cancel:
d%2F10+=+log%28%28I%5B1%5D%2FI%5B2%5D%29%29
Next we rewrite the equation in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to a%5Eq+=+p. Using this pattern (and the fact that the base of "log" is 10) we get:
10%5E%28%28d%2F10%29%29+=+I%5B1%5D%2FI%5B2%5D
Multiplying both sides by I%5B2%5D we get:
I%5B2%5D%2A10%5E%28%28d%2F10%29%29+=+I%5B1%5D
This equation says what we set out to find: The intensity of the louder sound, I%5B1%5D, is (equals) 10%5E%28%28d%2F10%29%29 times the intensity of the softer sound, I%5B2%5D.