Figure 6-37. Scale for Determining Logarithms
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T.O. 33B-1-1
6-73
6.7.4.5
Logarithms
.
The use of logarithms was covered in Section III. This review explains the use of Figure 6-37, the "Scale for
Determining Logarithms".
Figure 6-37. Scale for Determining Logarithms.
6.7.4.5.1
Since logarithms are used a great deal in the following section, a brief discussion of them is included here. A more
detailed treatment will be found in Paragraph 6.4.1.5 and some handbooks and intermediate algebra texts. Before
discussing logarithms, it will be necessary to define the term "power". The power of a number is the product obtained
when it is multiplied by itself a given number of times. Thus 10
^{3}
= 10 X 10 X 10 = 1000; 5
^{2}
= 5 X 5 = 25. In the first
example, 1,000 is said to be the third power of 10; in the second, 25 is the second power of 5, or 5 raised to the second
power. The figure
^{2}
is known as the exponent.
Fractional exponents
are used to denote roots.
6.7.4.5.1.1
For example: Negative exponents indicate
reciprocals
of powers. Thus: The common logarithm of a number is the
exponent or the power to which ten must be raised to give the number in question. For example, the logarithm of l00 is
2. The logarithm of 3l6 = 2.50, or log 3l6 = 2.50; the logarithm of l000 is 3, or log l000 = 3. It is also said that 1000 is
the antilogarithm of 3, or antilog 3 = 1000. Logarithms consist of two parts: a decimal that is always positive called the
mantissa; and an integer which may be positive or negative, called the characteristic. In the case of log 316 = 2.50, .50
is the mantissa and 2 is the characteristic. The mantissa may be found by reference to a table of logarithms, by the use
of a slide rule (D and L scales), or by reference to Figure 6-37. No matter what the location of the decimal point may
be, the logarithms of all numbers having the same figures in the same order have the same mantissa. The characteristic
of the logarithm is determined by the location of the decimal point in the number. If the number is greater than one,
the characteristic is positive and its value is one less than the number of digits to the left of the decimal point. If the
number is less than one (i.e., a decimal fraction), the characteristic is negative, and has a numerical value of one
greater than the number of zeros between the decimal point and the first integer. A negative characteristic of, say, 3, is
written 3 ... to indicate that only the characteristic is negative, or 7 . . . -10. From Figure 6-37 we see that the mantissa
of the logarithm of 20 is 0.30. The characteristic is 1.
log 20
=
1.30
log 40
=
1.60
log 80
=
1.90
log 160 =
2.20
log 200 =
2.30
log 2000 = 3.30
log 20000 = 4.30
log 0.2
=
1.30 or 9.30 - 10
log 0.02 =
2.30 or 8.30 - 10
The tabulation above illustrates a very important property of logarithms. Note that when a series of numbers increases
by a constant factor (e.g., the series 20, 40, 80, 160 or the series 20, 200, 2000, 20,000), their logarithms have a
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