By such processes as these the natural logarithm of any number can be found to any requisite degree of approximation. 362. In Art. 302 it was shown that In 10 is the reciprocal of the modulus of the system of logarithms whose base is 10. Hence the modulus of common logarithms is M=1/ln 10=1/2.302585... CHAPTER XXXVI. LOGARITHMIC COMPUTATION. 363. The logarithms used in all theoretical investigations are natural logarithms; but for the purpose of making approximate numerical calculations, for reasons that will shortly appear, logarithms to base 10 are always employed. On this account logarithms to base 10 are called common logarithms.* We have shown, in the preceding chapter, how natural logarithms, or logarithms to base e, can be found; and having constructed a table of natural logarithms, the logarithms to base 10 are obtained by multiplying the former by the modulus of the latter, that is, by the constant factor log e [Art. 291, VI.], whose numerical value 364. In what follows the logarithms must always be supposed to be common logarithms, and the base 10 need not be written. * Also Briggian logarithms in honour of Henry Briggs, of Oxford, who first constructed tables of such logarithms. If two numbers have the same figures, and therefore differ only in the position of the decimal point, the one must be the product of the other and some integral power of 10, and hence from Art. 297, III., the logarithms of the numbers will differ by an integer. Thus log 421.5 log 4.215 + log 100 2 + log 4.215. Again, knowing that log 3.30103, we have On account of the above property, common logarithms are always written with the decimal part positive. Thus log .03 is not written in the form -1.69897, but 2.30103, the minus sign referring only to the integral portion of the logarithm, and being written above the figure to which it refers. Definition. When a logarithm is so written that its decimal part is positive, the decimal part of the logarithm is called the mantissa, and the integral part the characteristic. 365. The characteristic of the logarithm of any number can be written down by inspection. First let the number be greater than 1, and let n be the number of figures in its integral part; then the number is clearly less than 10" but not less than 10"-1. The logarithm of the number is therefore between n and n-1; thus the logarithm is equal to n - 1+ a decimal. Thus the characteristic of the logarithm of any number greater than unity is one less than the number of figures in its integral part. For example, 235 is greater than 102 but less than 103. Hence log 235 = 2 + a decimal, so that the characteristic is 2. Next, let the number be less than 1. Express the number as a decimal, and let ʼn be the number of ciphers before its first significant figure. Then the number is less than 10-" and not less than 10-n-1. Hence, as the decimal part of the logarithm must be positive, the logarithm of the number will be - (n+1)+ a decimal fraction, the characteristic being − (n + 1). Thus, if a number less than unity be expressed as a decimal, the characteristic of its logarithm is negative and greater by one than the number of ciphers before the first significant figure. For example, .02 is greater than 10-2 but less than 10-1; hence log .02 is 2+ a decimal, the characteristic being - 2. Also .00042 is greater than 10-4 but less than 10-3; hence log .00042 is - 4+ a decimal, the characteristic being - 4. 366. Conversely, if we know the characteristic of the logarithm of any number whose digits form a certain sequence of figures, we know where to place the decimal point. For example, knowing that log 1.1467.0594498, we know that the number whose logarithm is 3.0594498 is 1146.7, and that the number whose logarithm is 4.0594498 is .00011467. 367. Tables are published which give the logarithms of all numbers from 1 to 99999 calculated to seven places of decimals: these are called "seven-figure" logarithms. For many purposes it is, however, sufficient to use fivefigure, or even four-figure, logarithms. In all Tables of logarithms the mantissæ only are given, for, as we have seen, the characteristics can always be written down by inspection. |