One unit of ( X ) is equal to ten times one-tenth of ( X ). We are at liberty to allow ( X ) to be a quantity of anything we wish. For example, let’s allow ( X ) to be a dollar bill:
1 dollar = ( 10 )( 1 tenth dollars )
1 tenth dollar = 1 dime
1 dollar = ( 10 )( dimes )
Before moving into the subject of decibel ( dB ) notation, it is important to review how exponential and logarithmic functions are related.
10^1 = ( 10 )( 1 ) = 10
10^2 = ( 10 )( 10 ) = 100
10^3 = ( 10 )( 10 )( 10 ) = 1,000
The 1, 2, and 3 exponents above answer the question of how many factors of 10 exist within some number. The logarithmic ( log ) function is the opposite process of the exponential function. Log functions answer the question of what exponent would be needed to raise the number 10 to a given value:
log 10 = 1
log 100 = 2
log 1,000 = 3
The input values in the log functions above could just have easily been ratios of related numbers:
( 100 / 10 ) = 10
( 1,000 / 10 ) = 100
( 10,000 / 10 ) = 1,000
log ( 100 / 10 ) = 1
log ( 1,000 / 10 ) = 2
log ( 10,000 / 10 ) = 3
In the examples above, the log function shows how much larger the numerator is compared to the denominator by factors of 10. For the sake of simplicity, larger multiples of 10 were used in the numerator, but this will not always be the case.
When dealing with sound, the numerical ratios of interest are comprised of initial and final intensity measurements of sound being emitted by a source. The initial intensity is denoted using the ( Io = ” I knot ” term ), and the final intensity measurement is denoted using ( If ):
log ( If / Io ) = y
If the perceived intensity of a sound source has increased 100 times, the log function will quantify this relationship as follows:
Io = X
If = 100X
log ( If / Io ) = log ( 100X / X )
log ( 100 ) = 2
The number 100 is two factors of 10 larger than the number 10.
Logarithmic functions using intensity ratios as input values are expressed in terms of Bels ( B ) and decibels ( dB ). The ” deci ” prefix denotes that ” one-tenth ” of some value must be expressed. Let’s first quantify the previous result in terms of Bels and subsequently convert to decibel ( dB ) notation:
B = log ( 100X / X )
B = log ( 100 )
B = 2
The 2 above can be rewritten in terms of a value that, when multiplied by the number 10, is an equivalent statement; one-tenth of 2, multiplied by 10, is 2 as well:
B = ( 10 )( 2 dB ) = 20 dB
Putting everything together, we have the following expression:
B = 10 log ( If / Io )
Whenever a decibel reading has been given, we must divide by 10 to see how many factors of 10 that ( If ) has increased ( or decreased ) relative to ( Io ):
20 dB
20 dB = ( 10 )( 2 dB )
( 20 dB / 10 dB ) = 2
10^2 = 100
Interestingly, the human ear doesn’t interpret sound in a linear fashion. For every factor of 10 that the sound intensity of a source increases ( or decreases ), the human temporal region of the brain interprets that a doubling of sound has occurred. A 20 dB increase in the sound intensity of a source would be interpreted in the following manner:
2^x = y
2^2 = A fourfold increase in perceived loudness has occurred.
P.S. Notice how B = 10 log ( 10 / 10 ) = B = 10 log ( 1 ) = 0, which indicates that no change in intensity has occurred ( Io = If ).
RENAISSANCE PHYSICS 