It is necessary to compute directly the logarithms of prime numbers only, in any system; for, according to 404, 3, the logarithm of any composite number may be obtained, by adding the logarithms of its factors. 413. We will now illustrate the use of formula (B), by computing the Naperian logarithms of 2, 4, 5, and 10. Make z=1; then nap. log. z = 0, and nap. log. (z + 1) = nap. log. 2; and since M1, we have Form a column of numbers consisting of and the quotients obtained by dividing by 32, or 9, continually; then dividing the first of these numbers by 1, the second by 3, the third by 5, and so on, we obtain the several terms of the series. Next make z = 4; then z + 1 = 5; and 2z + 1 = 9; and we 414. In order to compute common logarithms, we must first determine the modulus of the common system. From 411, equation (3), we have M= log. (1 + p) nap. log. (1 + p) In this equation, make 1 + p = 10, the base of the common system. Then we have the value of the modulus sought. Substituting this value in formula (B), we obtain the formula for common logarithms, as follows: .86858896 (2 z +1 + 3 ( 2 z+1)3 + 5 (2 z + 1 ) + 7(2z+1)2' To apply this formula, assume z = 10; then log. z = :1, and 2z + 1 = 21. 212 441 .04136138 ÷ 1 = .04136138 log. (z+1)= 1.04139268= log. 11. If we make z 99, then z +1= 100, and 2z+1= 199. In this case, the formula will give the logarithm of 99; for, log. (z+1) log. z = log. 100 log. 992-log. 99. 199.86858896 1992 39601 4364771= .00436477 11÷ 3 = 4 .00436481, sum of series. Thus we may compute logarithms with great facility, using the formula for prime numbers only. USE OF TABLES. 415. The following contracted tables will illustrate the principles of logarithms, and the methods of using the larger tables. The logarithms are taken in the common system. TABLE II.-LOGARITHMS OF LEADING NUMBERS WITHOUT INDICES. 100 C00000 000434 000868 001301 001734 002166 002598 003029 003461 003291 191 004321 604750 005181 005609 006038 006466 006894 007321 007748 008174 102 008600 009026 009451 009876 010300 010724 011147 011570 011993 012415 103 012837013259 013680 014100 014521 014940 015360 015779 016197 016616 104 017033 017451 017868 018284 018700 019116019532 019947 020361 020775 105 021189 021603 022016.022428 022841 023252 023664 024075 024486 024896 106 025306 025715 026125 026533 026942 027350 027757 028164 025571 028978 107 029384 029789 030195 030600 031004 031408 031812 032216 032619 035021 108 033424 033826 034227 034628 035029 035430 035830 036230 036629 037C28 109 037426037825 038223 038620 039017 039414 039811 040207 040602 040998 In table I, the logarithms are given, with indices, in columns adjacent to the columns of numbers. In table II, each figure in the row at the top may be annexed to any number in the left-hand column; the logarithm of any number thus formed, will be found at the right of the number in the column, and beneath the figure at the top. The proper index may be supplied in any case, according to the theory of logarithms. Thus, to obtain the logarithm of 1023 by this table, we find 102 in the left-hand column, and 3 in the top row; and opposite the former, and under the latter, we find 009876, the decimal part of the logarithm. Hence, log. 1023=3.009876. In like manner, we find log. 104.22.017868, log. .1078: == - 1.032619. CASE I. 416. To find the logarithms of numbers when their factors are in the tables. RULE.-Take out from the tables the logarithms of the factors, and find their sum; the result will be the logarithm required. 417. To find the logarithms of numbers intermediate between the numbers in the table. Since the logarithms in any table form a regular series, we may interpolate for intermediate logarithms, by the usual formula, If the logarithm of the given number is intermediate between the logarithms of table I, it will be necessary to take account of the first and second differences. But we may always employ table II, where the logarithms increase so slowly that two terms of the formula will give the result accurately. The first four figures of a number, counting from the left, will be called the four superior figures; and the others, the inferior figures. To apply the formula, a will represent that logarithm of the table which is next less than the required logarithm, and n will denote the inferior figures of the number, regarded as a decimal. Hence the following RULE.-Take out the logarithm of the four superior figures of the given number; multiply the difference between this logarithm and the next greater in the table, by the inferior places of the number, considered as a decimal; add this product to the former result, and the sum will be the logarithm required. |