Indices and logarithms

2007-04-01 2:33 am

The loudness, measured in decibels, is defined by the function

b = 10 log ( I / I o ) ,

where I is the intensity of the sound and I o is the minimum intensity detectable. How many times greater is the intensity of b1 = 105dB ( factory ) than the intensity of b2 = 80dB ( busy street ) ?

回答 (1)

2007-04-01 4:15 am

✔ 最佳答案

Given b = 10 log ( I / I o) where Io is a constant & b1 = 105, b2 = 80
Le the intensity of b1 = Ib1, the intensity of b2 = Ib2 , so that:
105 = 10 log ( Ib1 / Io ) .............(1)
80 = 10 log ( Ib2 / Io ) ...............(2)

(1) - (2):
105 - 80 = 10 log ( Ib1 / Io ) - 10 log ( Ib2 / Io )
25 = 10 [log ( Ib1 / Io ) - log ( Ib2 / Io )]
25/10 = log [( Ib1 / Io ) / ( Ib2 / Io )]
5/2 = log ( Ib1 / Ib2 )
Ib1 / Ib2 = 105/2

Ib1 / Ib2
= 102 + (1/2) = 102101/2 =100√(10)
~ 316.23 ( corrected to 2 decimal places)

Therefore, the intensity of b1 = 105dB ( factory ) is ~ 316.23 times greater than the intensity of b2 = 80dB ( busy street ).

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