1.2.3.4.5 +.... Reverting the series, we obtain so c=st 1.2 + 1 1 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. 411. In the general formula, (A), the quantity M, which depends upon the base, is called the modulus of the system. Thus, the 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, po pop* po (1) 2 5 M = p po (2) 2 3 5 Dividing (1) by (2), we obtain log. (1+p) (3) nap. log. (1+p) or, {nap. log. (1+P) XM = log. (1+p), , (4) 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, p på P- (1) 2 3 4 5 If in this equation we substitute -P for P, we shall have p p* po M (2) 2 3 4 5 If we subtract equation (2) from equation (1), observing that P 1 + -....). ...). log:(1+p)-log:(1-P) = log. we shall have log. (1+0), ...). (8) + 1+p pe po = 2M ( pt + + 3 5 7 1 1+p Assume P = ; whence we obtain 22+1 These values substituted in equation (3), give z+1 1-P +..). 1 1 + (4) 2z+1 3(2x+1) 5(2x+1) 7(2x+1)' The first member of this equation is equivalent to log. (2+1), log. 2. Hence, finally, we have og (z+1)-log.z= 1 1 1 1 2M + + + +. (B) 3(2x+1) 5(2z+1) 7(2x+1)' 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. (2+1), or log. 2. Next make z = 2; then the formula will give the value of log. (2+1), or log. 3 ; and so on. (22 +1 + ..). nap. log. 2 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. 2 = 0, and nap. log. (z+1)= nap log. 2; and since M=1, we have 1 1 1 2 + 3.30 + + 5:36 7.37 We first form a column of numbers, by dividing by 3', 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. 32 161609 13064 1129 103 10 1 .69314718 = nap. log. 2. 2 Whence, by (404, 5), 1.38629436 = nap. log. 4. Next make z=4 ; then z+1=5; and 2z+1=9; and we have 1 1 1 1 nap. log. 5=2 + + + +...+nap. log. 4 7.97 91449 677 6 . 1.9 M м M= Το .22314354 Add nap. log. 4 = 1.38629436 1.60943790 = nap. log. 5. Add nap. log. 2 = .69314718 Whence, by (404, 3), 2.30258508 = nap. log. 10. 414. In order to compute common logarithms, we must first determine the modulus of the common system. From (411), equation (3), we have log.(1+p) nap. log:(1+p) In this equation, make 1+p = 10, the base of the common system. Then we have 1 (1) 2.30258508 the value of the modulus sought. Substituting this value in formula (B), we obtain the formula for common logarithms, as follows: log. (2+1)-log.z = 1 + (0) 1) -1)' To apply this formula, assume z = 10; then 21 | .86858896 3126 4 .04139268, sum of series. Add log. z = 1.0 log. (2+1) = 1.04139268 = log. 11. If we make z=99, then z+1= 100, and 22+1 = 199. In this case, the formula will give the logarithm of 99; for, log. (z+1)-log. 2= log. 100—log. 99 = 2— log. 99. 199 .86858896 1999 = 39601 436477 ; 1= .00436477 11 ; 3 = 4 .00436481, sum of serios. 21 log. 99. Therefore, we have 2-log. 99 = .00436181, whence, 1.99563519 = Subtract log. 11 = 1.04139268 .95424251 = log. 9 And by (404, 6), log. 9 = 47712126 = log. 3. 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 princi. ples of logarithms, and the methods of using the larger tables. The logarithms are taken in the common system. |