T.O. 33B-1-16-736.7.4.5 Logarithms.The use of logarithms was covered in Section III. This review explains the use of Figure 6-37, the "Scale forDetermining Logarithms".Figure 6-37. Scale for Determining Logarithms.6.7.4.5.1Since logarithms are used a great deal in the following section, a brief discussion of them is included here. A moredetailed treatment will be found in Paragraph 6.4.1.5 and some handbooks and intermediate algebra texts. Beforediscussing logarithms, it will be necessary to define the term "power". The power of a number is the product obtainedwhen 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 firstexample, 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 secondpower. The figure^{2} is known as the exponent. Fractional exponents are used to denote roots.6.7.4.5.1.1For example: Negative exponents indicate reciprocals of powers. Thus: The common logarithm of a number is theexponent or the power to which ten must be raised to give the number in question. For example, the logarithm of l00 is2. 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 isthe antilogarithm of 3, or antilog 3 = 1000. Logarithms consist of two parts: a decimal that is always positive called themantissa; and an integer which may be positive or negative, called the characteristic. In the case of log 316 = 2.50, .50is the mantissa and 2 is the characteristic. The mantissa may be found by reference to a table of logarithms, by the useof a slide rule (D and L scales), or by reference to Figure 6-37. No matter what the location of the decimal point maybe, the logarithms of all numbers having the same figures in the same order have the same mantissa. The characteristicof 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 thenumber is less than one (i.e., a decimal fraction), the characteristic is negative, and has a numerical value of onegreater than the number of zeros between the decimal point and the first integer. A negative characteristic of, say, 3, iswritten 3 ... to indicate that only the characteristic is negative, or 7 . . . -10. From Figure 6-37 we see that the mantissaof the logarithm of 20 is 0.30. The characteristic is 1.log 20 = 1.30log 40 = 1.60log 80 = 1.90log 160 = 2.20log 200 = 2.30log 2000 = 3.30log 20000 = 4.30log 0.2= 1.30 or 9.30 - 10log 0.02 = 2.30 or 8.30 - 10The tabulation above illustrates a very important property of logarithms. Note that when a series of numbers increasesby 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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