39. 7 log (a + b) — 3 log (a — b) + 1⁄2 log x 40. log (ax - b) — § log (cx — d) + 3 log (mx 41. log (a2 + b2) — § [log (a + b) + log (a − b)]. 42. 2 log (xy)- log (x-xy+y) - log (x + y). y COMMON SYSTEM OF LOGARITHMS 562. It is possible to have any number of systems of logarithms but common usage has adopted only two, viz., the Napierian system and the Common system. The Napierian system, named after its inventor, John Napier, is used in theoretical investigations; its base is The Common system of logarithms is used in numerical calculations; the base in this system is 10. NOTE.-The advantage due to the use of the base 10 will be seen in the rules for the Characteristics which immediately follow and the Characteristics given in the next paragraph. It follows from this table that the logarithms of numbers greater than 1 consist of two parts, an integral part and a decimal part. The integral part is called the Characteristic and the decimal part the Mantissa. It follows also from the preceding table that if a number is expressed by one digit the characteristic of its logarithm (e. g., log 7) is 0; if it is expressed by two digits the characteristic of its logarithm (e. g., log 89) is 1; if it is expressed by three digits the characteristic of its logarithm (e. g., log 749) is 2; and so on, the characteristic being one less than the number of digits in the number. Thus, the characteristic of log 11749 is 4, of log 6748.63 is 3. Therefore the characteristic of the common logarithm of any number greater than 1 can be written down by the following rule. 563. RULE I.—The characteristic of the logarithm of a number greater than unity is positive and is equal to the number of digits in its integral part less one. 564. Similarly, if the number is less than 1 and greater than 0, From an inspection of this table it is clear that, in the common system, the logarithm of any number between 10; 1 and .1 is some number between 0 and 8. decimal 10; 3, 7. decimal - 10; and so on. .01 and .001 is some number between 2 and e. g., log .008 = - 3+ decimal In other words, the logarithm of any decimal with no zero between its point and first figure, is equal to 9 plus some decimal, minus 10; the logarithm of any decimal with one zero between its point and first figure, is equal to 8 plus some decimal, minus 10; the logarithm of any decimal with two zeros between its point and first figure, is equal to 7 plus some decimal, minus 10; and so forth. In General. Let D be a decimal with n zeros immediately after the decimal point; then Therefore the characteristic of any number less than 1 and greater than zero is determined by the following rule. 565. RULE II.—If the number is less than 1 the characteristic is found by subtracting the number of zeros between the decimal point and the first significant figure from 9; writing -10 after the mantissa. E. g., the characteristic of logarithm .006797. decimal with 10 written after the mantissa; and the characteristic of logarithm .3796 9. decimal with — 10 written after the mantissa. In practice it is customary to omit -10 after the mantissa; it is however a part of the logarithm, and should be allowed for and subjected to exactly the same operations as the rest of the logarithm. Beginners will find it useful to write 10 in all cases, and in many problems it can not well be omitted. NOTE.-Many writers, in using logarithms less than 1, combine the two parts of the characteristic and write the result as a negative characteristic before a positive mantissa. Thus, instead of the logarithm 7.573963 -10, the student will frequently find 3.573963, a minus sign being written over the characteristic to show that it alone is negative, the mantissa being always positive. A well-founded objection to this notation is that it is inconvenient to use numbers partly positive and partly negative. USE OF THE TABLE 566. Calculations have been made for the common logarithms of all integers from 1 to 200,000, and the results tabulated. For general use, tables of logarithms give six decimal places, but those used especially for astronomical and mathematical calculations give seven or more decimal places. In the examples we use the common table, which gives the mantissas of the logarithms of all integers from 1 to 10,000 calculated to six decimal places. For convenience, the logarithms of integers from 1 to 100 are given on the first page, but the same mantissas are to be found in the rest of the table. 1. The characteristic of a logarithm can be written according to the rules of 563, 565, and the mantissa looked up in the tables. 2. The mantissas of the logarithms of all numbers which have the same sequence of figures is the same; so only the mantissas of integers are given in the table. For, let N be a number of any sequence of figures, then log 10(NX 10")=log 10N+log 1010"=log 10N+n, log 10(N÷10")=log 10N-log 1010"-log 10-n. Thus only the characteristic of the logarithm is affected. EXAMPLES. 1. Given log 10 3296.78 = 3.518090, then log 10329678=log 10 (3296.78 x 102) = log 103296.78+2 =3.518090+2=5.518090. 2. Log 103.29678=log 10(3296.78÷103) =logo 3296.78-3-3.518090-3=0.518090. 10 3. Log 10.00329678=3.518090-6=7.518090–10. That is, in the common system of logarithms, if the logarithm of any number is known, we can immediately determine the logarithm of the product or the quotient of that number by any power of 10, by adding n to or subtracting n from the characteristic, according as the number is multiplied by 10" or divided by 10": 567. To find the Logarithm of any Number Consisting of Four Figures. Find, in the column headed N, the first three figures of the given number. The mantissa required will be found at the intersection of the horizontal line through these three figures and the vertical column headed by the fourth figure. If only the last four figures are found, the first two figures may be obtained in the same vertical column from the first mantissa above consisting of six figures. Prefix the proper characteristic (2563 or 565). • For example, log 579.8 2.763278, log.006847 =7.835500-10, log 9899 3.995591. To find the mantissa of a logarithm of a number consisting of two figures, use the first page of the table; for a number consisting of three figures, look in the column headed N and take the mantissa from the column headed 0. 568. To find the Logarithm of a Number of More Than Four Figures. - For example, find the logarithm of 356.478. From the table, the mantissa of 3564 = = .551938 mantissa of 3565 = .552060 Hence, the change in the mantissa correspond ing to a change of one unit in the number } = .000122; therefore the change in the mantissa due to a change of .78 units in the number is .78 × .000122 = .000095 correct to the sixth decimal place. Hence, mantissa of 3564 = .551938 Correction for .78 is .78 x .000122 = .000095 log 356.478 = 2.552033 NOTE. In making the correction in the mantissa corresponding to .78 increase in the number, it was assumed that the differences in logarithms are proportional to the differences of the corresponding numbers, which is not exactly true but is sufficiently accurate for practical purposes. |