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In order to study the decibel, you must first review
something about the logarithm. Logarithms
are simply exponents and you know what an exponent is; in the expression 102 = 100,
the 2 is an exponent.
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exponent
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(1) In the expression 102 = 100,
the 2 is an .
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exponent
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(2) We have said that a logarithm is an .
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logarithm
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(3) Therefore, the number 2 in the expression 102 = 100
is both an exponent and a .
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The exponent or logarithm above the number 10 tells
us how many times to use the ten in multiplication. Thus, 102 means the same as
10 * 10 or 100.
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multiplication
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(4) The expression 103 means
use the number 10 three times in .
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ten multiplication
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(5) The expression 104 means
use the number
four times in .
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6
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(6) The expression 10___ means
use the number 10 six times in multiplication.
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10 10 10
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(7) The expression 103 means the same as
___ * ___ * ___, or 1,000.
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10 * 10 * 10 * 10 * 10 (or 100,000)
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(8) The expression 105 means
the same as
.
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logarithm
multiplication
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(9) In the expression 105 the number 10 is
called the base, while the number 5 is the exponent or
which tells you how many times to use
the base in .
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4
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(10) In the expression 43 and
103 the exponents are the same (the number three) but the bases
are _____ and 10 respectively.
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4 4 4
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(11) If 103 means
10 * 10 * 10, then 43 must mean
_____ * ______ * _____.
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10 6
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(12) In the expression 106 we mean use the
number _____ (how many) _____ times in multiplication.
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1,000,000
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(13) Thus, 106 = (whole
number) __________.
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10,000
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(14) While 104 = (whole
number) __________.
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3
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(15) The number 1,000 can be expressed as
10__.
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4 10
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(16) The logarithm in the expression 104
is the number _____, while the base is the number ______.
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100
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(17) 102 = _______.
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2
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(18) The logarithm of 100 is _____.
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3
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(19) 103 = 1,000,
which can be expressed as the log of 1,000 is ____.
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4
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(20) The log of 10,000 is ______.
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You may have noticed that when the base is 10, the exponent
tells you how many zeroes appear after the number 1. Thus, 102 is the number 1
followed by two zeroes, or 100. The
expression 104 is the number 1 followed by four zeroes or 10,000.
In dealing with decibels, the base we shall be
concerned with will always be 10. When we ask, "What is the logarithm of
100," for example, we mean, "What is the logarithm of 100 to the base 10?"
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9
1,000,000,000
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(21) 109 is the number 1
followed by ______ zeroes, or the whole number _______________.
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10
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(22) It follows then that 101 = _______
and 100 = 1.
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It is very important to the concept of the decibel to
remember that the logarithm of 1 is zero.
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1 0 0
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(23) 100 =________. 10___ = 1. The log of 1 is ________.
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logarithm
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(24) The ______ of 1 is zero.
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3
3
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(25) The log of 1,000 is _______. This is the same as 10___ = 1,000.
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2
2
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(26) The log of 100 is ______. This is the same as 10___ = 100.
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1
1
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(27) The log of 10 is ________. This is the same as 10___ = 10.
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0
0
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(28) The log of 1 is _____. This is the same as 10___ = 1.
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1
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(29) The log of ______ is zero.
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We must also review a bit about ratios. If we divide a number by itself, as in 198/198,
we get a ratio of 1. If we divide
3,456 by 3,456, we still get a ratio of 1.
If we divide 0.05 by 0.05, we still get a ratio of 1.
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1
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(30) If we divide 20 by 20, we get a ratio
of _________.
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1
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(31) Regardless of the magnitude of the numbers
chosen, any number divided by itself equals _______.
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10 1
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(32)  = _______ or 10___.
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100 2
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(33)  = _______ or 10___.
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1,000 3
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(34)  = _______ or 10___.
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1 0
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(35)  = _______ or 10___.
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THE DECIBEL IN MEASUREMENTS FROM A POWER REFERENCE
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Study the following equation:
#
dB IL = 10 * log 
WO = watt/cm2
(power) output, and
WR = watt/cm2 (power)
reference.
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output
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(36) WO = watt/cm2 power ______,
and
WR = watt/cm2 power
reference.
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reference
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(37) WO = watt/cm2 power output
while
WR = watts /cm2 power _______.
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watt/cm2 power output
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(38) WO = ________.
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watt/cm2 power reference
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(39) WR = __________________.
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To solve the equation, one must first find the
numerical value of  . This is a ratio.
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1,000
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(40) If WO = 100 and
WR = 0.1, then the ratio = ________.
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1
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(41) If WO = 1,000
and WR = 1,000, then the ratio = ______.
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If your mathematics is sufficiently advanced to
permit you to answer Item 42, you may then skip to Item 54.
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104
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(42) If WO = 10-12 and WR = 10-16,
then the ratio of
 is 10,000 or
_______.
If you could not answer Item 42, you must read the next section before you
can go on; otherwise, go to Item 54
The numbers such as the 3 in 103 which you
saw were called logarithms, or exponents, and told you how many times to use the
base 10 as a factor in multiplication.
The exponents with a minus
sign before them, such as the -3 in 10-3 tell you how many times
to use the base in division, like
this:
If 103 = 10 * 10 * 10 then 10-3 = 1/(10 * 10 * 10)
or 1/1,000 or 0.001.
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 =  = 0.0001
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(43) If 104 means
10 * 10 * 10 * 10, then 10-4 means
_________________.
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These exponents are basic to what is called
scientific notation, a language which will be clarified in a later
program. If 104 means
10,000, then 10-4 means  .
Now you recall from the footnote on page 3 that when
you multiply numbers such as 104 * 108, you add the exponents like this: 104+8, or 1012. If you were to divide 108 by 103
as in this expression:  , you subtract the
exponents like this: 108-3,
or 105.
Try these:
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1010
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(44) 107 * 103 = _______
If you got that correct, you successfully multiplied
10 million by 1,000 to get an answer of 10 billion or 10,000,000,000, which is
more simply expressed as 1010
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108
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(45) 106 * 102 = ________.
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105
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(46) 108/103 = _________.
Now the next is a critical item:
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10-3
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(47)  = ________.
That was a difficult frame. It teaches that the subtraction of the exponents
requires that the lower figure (the denominator) be subtracted from the upper
figure (the numerator) regardless of
which number is larger.
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10-2
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(48)  = _______.
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10-13
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(49)  = _______.
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10-1
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(50)  = _______.
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101
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(51)  = _______.
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If you correctly answered Item 51, you knew that when
you subtract numbers which are preceded by a minus sign, you essentially do
this:
(-4) - (-5) = (-4) + 5 = 1.
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104
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(52)  = _________.
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104
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(53)  = _________.
If you answered this item correctly, you have learned
Item 42, which had stopped you before.
You may recall the item said:
If WO = 10-12 and WR = 10-16,
then the ratio is 104 or 10,000.
If you did not answer Item 53 correctly, go back to
Item 43 and begin again.
If you
missed Item 53 a second time, stop working on the program and see your
mentor. Be sure this mathematical
hurdle is cleared before you try to go any further.
You recall we said that  =  , or 10,000.
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10
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(54) Then we multiply the log of 10,000 by _______ as in 10 * log
10,000.
Remember that when we use numbers such as 104
or 102, the exponents are the logarithms; but if we used the
numbers 10,000 or 100, we would have to find their logarithms before we could
multiply them by 10.
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40
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(55) Thus, when WO = 10-12
and WR = 10-16, the decibel output with
regard to the reference is 10 * log  or ________ dB IL.
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130
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(56) If WO is 10-3 and the reference is 10-16
watt/cm2, then the decibel output with regard to the reference is
_________ dB IL.
Physicists and engineers have settled on an Intensity
Level Reference of 10-16 watt/cm2 when we talk about
references from which to make Intensity Level measurements.
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1
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(57) If WO = 10-16 and WR also = 10-16
then the ratio of WO over WR is ________.
We are now on the verge of one of the most critical
parts of this program.
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0
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(58) When WO = WR
as in the case of WO = 10-16 and WR = 10-16,
the ratio of the reference to the output is 1; but the logarithm of 1 is zero,
therefore, the decibel output with regard to the reference is _________ dB
IL.
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0
0
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(59) If the ratio between WO
and WR equals 1, and we know that the logarithm of one equals ______,
then the decibel output with regard to the reference is also ______ dB IL.
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equal
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Thus, 0 dB does not mean silence, or absence of sound
or absence of power, nor does it mean very faint sound or power, either. It simply means that the power output of
the system is exactly _______ to the reference from which the decibel
measurement is started.
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1
0 0 0
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(60) When WO = WR,
the ratio is _______, the logarithm of 1 is _____, and
10 * _____ = _____ dB IL.
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power output
1
0
0
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(61) For example, if we chose as our
reference point (or WR) the value 100 watts/cm2, and WO
or (2 words) ________ ________ were
also 100 watts/cm2, the ratio of  would still equal
_______, the logarithm of that ratio would still be _______, and the
resultant decibel output with regard to
this new and different reference would still be _______ dB IL.
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Thus, either 10-16 watt/cm2 or
100 watts/cm2 can equal 0 dB, if they are chosen as references
from which to make other measurements.
It is quite permissible to choose any reference point from which to
make a dB measurement. In fact,
strictly speaking, every
decibel measurement is a decibel
difference from 0 or decibel difference with regard to the reference from
which the measurement was made.
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reference
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(62) The word decibel alone implies no fixed dimension of its own since the
_______ from which it is measured can be any value the experimenter chooses.
It is critical, therefore, to know the references from
which various decibel measurements are made.
The most common reference for measuring acoustic intensity
differences, when the variable is power,
is 10-16 watt/cm2. Decibels
described by the equation
10 * log WO/WR are expressed as dB IL or deciBels
Intensity
Level.
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70
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(63) If you must calculate the number of
decibels a 10-9 watt/cm2 power will generate, the
equation to use is: ________, and the dB IL re 10-16 watt/cm2
is __________.
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10-16
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(64) The most common and most likely
reference point from which this measurement will be made is
________ watt/cm2.
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30
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(65) However, if the reference were 10-12
watt/cm2, a 10-9 watt/cm2 signal will be
only ________ dB IL.
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reference
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(66) All that is required of the reporter
in describing his decibel is that he always
specify the _________.
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10-16
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(67) When you see the phrase dB IL or dB Intensity
Level,
this usually means that the reference was ________ watt/cm2.
Henceforth, let us assume a reference of
10-16 watt/cm2 equals 0 dB IL (or Intensity
Level).
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120
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(68) If the power output is 10-4 watt/cm2,
dB IL = _________.
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10
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(69) When WO = 10-15,
dB IL = _________.
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20
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(70) When WO = 10-14,
dB IL = _________.
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30
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(71) When WO = 10-13,
dB IL = _________.
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-20 (that is
right…minus 20 dB IL)
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(72) When WO = 10-18,
dB IL = _________.
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Notice that as the power is multiplied by 10, the dB output simply increases additively by units of 10. Thus, the power required to move from 0 dB
to 60 dB is not sixty units greater
than 10-16 watt/cm2, but 106 or 1,000,000
times greater than 10-16 watt/cm2. For your own use, construct a table like
this:
Power Measurements
 where WR = 10-16
watt/cm2
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0.1 -1 -10
10,000 4 40
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watt/cm2 (WO/WR)
(Wo) output Ratio Log dB IL
10-18 0.01 -2 -20
10-17 ___ ___ ___
10-16 1 0 0
10-15 10 1 10
10-14 100 2 20
10-13 1,000 3 30
10-12 ___ ___ ___
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THE DECIBEL IN MEASUREMENTS FROM A PRESSURE
REFERENCE
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In acoustics we make pressure measurements more often
than power measurements so we should know how to convert powers to
pressures. Scientists have known for
many years that powers (watts) and pressures (mPa) have a special relationship. Sound
pressure ratios are usually proportional to the square root of
corresponding power ratios, or power: pressure2.
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base
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(73) The exponent is a number which tells
you how many times to use the ________ in multiplication.
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10
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(74) If we start with dB
(power) = _____ * log  , and we say that to make power figures proportional to
pressure, we must square the pressure
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dB = 10*log
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where PO
is now a pressure output
where PR is now a pressure
reference
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When we square a number we multiply its logarithm by
2, and we can rewrite the equation in this way:
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dB
SPL = 10 * 2 * log
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therefore
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20
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(75) # dB SPL = * log |