EXAMPLES. 11 1. Find the eighth term of (3a2-6-1)". In this case, r = 8, n = 11; hence, the eighth term = 330(81a2) (- -") = - 26730a2b ̄1, Ans. Note. If the second term of the binomial is negative, it is convenient to enclose the term, sign and all, in a parenthesis, as shown in Ex. 1. Or, the absolute value of the term may be found by the formula, and the sign determined in accordance with the principle that the odd terms of the expansion are positive, and the even terms negative. 11. Ninth term of (m + m). 12 2 XXX. LOGARITHMS. 334. Every positive number may be expressed, exactly or approximately, as a power of 10; thus, 100 = 102; 13 = 101.1189...; etc. When thus expressed, the corresponding exponent is called its Logarithm to the base 10; thus, 2 is the logarithm of 100 to the base 10, a relation which is written log10 100 = 2, or simply log 100 = 2. And, in general, if 102 = m, then x = log m. 335. Any positive number except unity may be taken as the base of a system of logarithms; thus, if a = m, x is the logarithm of m to the base a. Logarithms to the base 10 are called Common Logarithms, and are the only ones used for numerical computations. If no base is expressed, the base 10 is understood. 336. By Arts. 220 and 221, we have Note. The second form of the results for log. 1, log.01, etc., is preferable in practice. 337. It is evident from the preceding article that the logarithm of a number greater than 1 is positive, and the logarithm of a number less than 1, and greater than 0, is negative. 338. If a number is not an exact power of 10, its common logarithm can only be expressed approximately; the integral part of the logarithm is called the characteristic, and the decimal part the mantissa. In this case the characteristic is 1, and the mantissa is. 1139. 339. It is evident from the first column of Art. 336 that the logarithm of any number between 1 and 10 is equal to O plus a decimal; Hence, the characteristic of the logarithm of a number, with one figure to the left of its decimal point, is 0; with two figures to the left of the decimal point, is 1; with three figures to the left of the decimal point, is 2; etc. 340. Similarly, from the second column of Art. 336, the logarithm of a decimal between 1 and.1 is equal to 9 plus a decimal 10; 10; :01 and .001 is equal to 7 plus a decimal 10; etc. Hence, the characteristic of the logarithm of a decimal, with no ciphers between its decimal point and first significant figure, is 9, with 10 after the mantissa; of a decimal with one cipher between its point and first figure, is 8, with after the mantissa; of a decimal with two ciphers between its point and first figure, is 7, with 10 after the mantissa; 10 etc. 341. For reasons which will be given hereafter, only the mantissa of the logarithm is given in the table; the charac teristic must be supplied by the reader. The rules for characteristic are based on Arts. 339 and 340. I. If the number is greater than 1, the characteristic is one less than the number of places to the left of the decimal point. II. If the number is less than 1, subtract the number of ciphers between the decimal point and first significant figure from 9, writing 10 after the mantissa. Thus, characteristic of log 906328.5 = 5; characteristic of log .00702 = 7, with -10 after the mantissa. Note. Some writers, in dealing with the characteristics of the logarithms of numbers less than 1, combine the two portions of the characteristic, writing the result as a negative characteristic before the mantissa. Thus, instead of 7.6036-10, the student will frequently find 3.6036; a minus sign being written over the characteristic to denote that it alone is negative, the mantissa being always positive. PROPERTIES OF LOGARITHMS. 342. The logarithm of a product is equal to the sum of the logarithms of its factors. Multiplying, 10 × 10 = mn, or 10x + y = mn. Substituting the values of x and y, log mn = logm+logn. In a similar manner the theorem may be proved for the product of three or more factors. 343. By aid of the theorem of Art. 342, the logarithm of any composite number may be found when the logarithms of its factors are known. 1. Given log 2 = .3010, log 3 = .4771; find log 72. log 72 = log(2×2×2×3×3) = log 2 + log 2+ log 2+ log 3 + log 3 = 3 x log 2 + 2 x log 3 = .9030+.9542 =1.8572, Ans. EXAMPLES. Given log 2 = .3010, log 3 = .4771, log 5 = .6990, log 7 = .8451; find the values of the following: 344. The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. |