Use the exponential form of the function L = 10 log(II0) to find the intensity of a sound if its decibel reading is 40. Explain what this means.

Question

Use the exponential form of the function L = 10 log(II0) to find the intensity of a sound if its decibel reading is 40. Explain what this means.

Answer 1

The intensity of a sound in decibels (dB) is typically defined as:

 

L = 10log(I/I_o)   [I think this is what is meant by what looks like 10log(110) in the introduction above]

 

where L = intensity in dB, I = sound intensity in watts/m², and I_o is the standard threshold for human hearing, typically taken as 10^-12 watts/m².  These values for sound intensity represent power per unit area at a given location.

 

This logarithmic definition is often used because the range of sounds in human hearing can be many, many powers of 10, which would be challenging to graph in detail on a single plot (and also require a lot of scientific notation to write).  The effect of the logartihmic measure is compress these powers of ten down to a much smaller scale, since each power of 10 only increases log by a factor of 1.

 

This is because logarithms are the inverse function of taking a base to a power (exponentiation).  As an equation, this idea can be written:

 

log(10^x) = x   (base-10 logarithms "undo" using a number as an exponent power of 10; just as taking the square root "undoes" squaring, for example)

 

Thus, log(100) = log(10²) = 2, and log(1000) = log(10³) = 3.  So even though the argument of the log function increased from 100 to 1000 (a factor of 10), the value of the log function only went up by 1.  This is the "scale compression" effect of the log function.

 

So, for this problem, we are told that the decibel reading is 40.  Thus, using the first equation above, we can write:

 

40 = 10log(I/10^-12) -->  4 = log(I/10^-12)

 

To "undo" log, we raise 10 to the power of both sides:

 

10⁴ = 10^{log(I/10^-12)} = I/10^-12

 

So then,

 

10⁴ = I/10^-12 and I = 10^-8 watts/m2.  This is the intensity of the sound.

 

This means that the sound is 10⁴ (or 10,000) times more intense than the threshold for human hearing.