c = 1+ + + +... + By taking 12 terms of this series, we find the approximate value of c to be 1.7182818. But the base is 1+c; hence, adding 1 to the result, and representing the sum by e, the usual symbol for the Naperian base, we have e = 2.7182818, which is the base of the Naperian system. Thus, the 411. In the general formula, (4), the quantity M, which depends upon the base, is called the modulus of the system. modulus of the Naperian system is unity. Let us here designate Naperian logarithms by nap. log., and logarithms in any other system by log., simply. Then, :) (1) (2) where M is the modulus of the system in which the logarithm of the second member is taken. Hence, The modulus of any particular system is the constant multiplier which will convert Naperian logarithms into the logarithms of that system. 412. Formula (A) can be employed for the computation of logarithms, only when p is less than unity; for if p be greater than unity, the series will be diverging. The series, however, may be transformed into another which will be always converging. Let us resume the logarithmic series, If we subtract equation (2) from equation (1), observing that log.(1+p)—log.(1—p) = log. (1+2), These values substituted in equation (3), give = log. (2+1) = 1 2M ·). (4) 1 + 7(2%+1)' +..). (2z+1 +3(2z+1)•+5(2x+1)* The first member of this equation is equivalent to log. (z+1) log. z. Hence, finally, we have 10g.(z+1)-log.z= 1 2 M 2z+1 + 1 1 1 + ..). (B) + This series is rapidly converging, and may be employed with facility for the computation of logarithms, in the Naperian, or in the common system. To commence the construction of a table, first make z = 1; then log. z = 0, and the formula will give the value of log. (z+1), or log. 2. Next make z = 2; then the formula will give the value of log. (z+1), or log. 3; and so on. It is necessary to compute directly the logarithms of prime num bers only, in any system; for, according to (404, 3), the logarithm of any composite number may be obtained, by adding the logarithms of its several factors. 413. We will now illustrate the use of formula (B), by com puting 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 M=1, we have We first form a column of numbers, by dividing by 32, or 9, continually; then dividing the first of these members by 1, the second by 3, the third by 5, and so on, we obtain the several terms of the series. Whence, by (404, 5), .69314718 nap. log. 2. 2 1.38629436 = nap. log. 4. 4; then z+1=5; and 2x+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 M= 1 2.30258508 = .43429448, (1) the value of the modulus sought. Substituting this value in formula (B), we obtain the formula for common logarithms, as follows: 86858896 log. (z+1)-log. z = 1 1 1 22 + 1 + 8 (22 + 1) + + 5 (22 + 1)• + 7 (22 +1)2+...). (0) 2z To apply this formula, assume z = 10; then log. z = 1, and 2x+1 = 21. log. (z+1)= 1.04139268 = log. 11. If we make z = 99, then z+1= 100, and 2x+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. |