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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;

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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.)

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Cor. 3.

A =
la la – 1) – 1 (a – 1)2 + } (a – 1): -...

-) *a ..

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DEF. The quantity A, or

lea

, by which the logarithms calculated to the base e are multiplied to give the logarithms

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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.

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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.)

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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

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lex

= m. (2m – 1).....

(vii.)

16. Having given Jex, to find l. (x +z), z being small when compared with x.

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1. (x + x) = ]q«.(1+
= 120 + 12 (1

1+ Expanding 12 (1+) by (vić), we have

(+)-1

+

1

(since

], 2 ir +%

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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.

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17. Having given the Napierian logarithms of two successive numbers, X - 1 and x, to find that of the number next following

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= 21«« – 12(x – 1) +1,(1-5).
Expanding 12 (1-) by (vi.) we have,
(1-a

1 (1-5) 1. (3+1)2.L»–1. (w-1)-2-4-, tee-13*+...}. (X.)

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1

[since

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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.

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=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

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or 1+p1: (x + x) + P2 : (x + x) + ... + Pn• (x + x)" + ...

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and equating the coefficients of the terms involving xx, xx,

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we have

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Now pı = (a – 1) - 4. (a – 1)2 + š. (a – 1): – ...

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