Cor. 1. Since A is a constant quantity, entirely dependent on the value of a, let e be the quantity whose value is such that e A = a; 1 We shall hereafter find the value of e, which is a constant quantity 2.7182818.... Logarithms to the base e are called Napierian, from Napier, the inventor of logarithms, who adopted this base because logarithms to it are more easily calculated than those to any other base; as is evident from comparing the series (ii.) and (iii.) Cor. 2. la = le{1+ (a – 1)}; which by (iii.) is = (a – 1) - } (a – 1)* + } (a – 1) - ............(iv.) 1 Cor. 3. A = -) *a .. 1 DEF. The quantity A, or lea , by which the logarithms calculated to the base e are multiplied to give the logarithms 1 to the base a, [for lan leN, Art. 3.] is called the Tea modulus of the system whose base is a. 15. We shall next investigate some rapidly converging formulæ for the calculation of logarithms. If x be a little greater than 1, this series converges very rapidly. Again, lex = ), (mm)" = m = m. leem 1 1 = m. {(um – 1)-1. (ivm – 1) +. ... }, by (iv.) and, (x being greater than 1), by assuming m of sufficient mag 1 nitude, arm may be made to differ from 1 by any definite quantity, however small the quantity may be; in which case, the succeeding terms of the series may be neglected as being of inconsiderable magnitude with respect to the first, and we have 1 lex = m. (2m – 1)..... (vii.) 16. Having given Jex, to find l. (x +z), z being small when compared with x. 1. (x + x) = ]q«.(1+ 1+ Expanding 12 (1+) by (vić), we have (+)-1 + 1 (since ], 2 ir +% 1 1 1 1 1.(1+x)=, +2 + + 2 x + 1 3.(2x + 1)3'5 (2x + 1) which is useful in computing le (1 + x) from lex, particularly when x is large. U 17. Having given the Napierian logarithms of two successive numbers, X - 1 and x, to find that of the number next following = 21«« – 12(x – 1) +1,(1-5). 1 (1-5) 1. (3+1)2.L»–1. (w-1)-2-4-, tee-13*+...}. (X.) 1 1 [since ], +1 18. To expand a' in a series ascending by powers of x; i.e. to expand the number in a series ascending by powers of the logarithm. =1+x.(a – 1) +w." -1.(a – 1)*+x. (x (x - 1) (x - 2) .(a – 1)*+... 2 2 3 ... let the coefficient of X, which is (a-1)-1. (a1)2+}. (a – 1)– .. be represented by Pi, and let the coefficients of w?, 203, ... be represented by P2, P3, ...; then we have |