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Expanding the second member by formula (19) of Art. 45, by making n=k, x=hA, we have

(14k)(1-2) b=1+(hA)+(1-K)

(hA)*+ 2

2.3. k3

(hA)* + &c. (38) Neglecting k, 2k, &c. in the factors 14k, 1—2k, &c. this equation becomes

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h and restoring x=1


b=1+xA+ **AP+***A®+ &c. (40) which is the required formula, expressing any number b in terms of its logarithm w and of A the reciprocal of the modulus of the system. If we substitute for x and A their values


x = log. b, A=
formula (40) becomes
1 1

1 log. 6/4

M 2.3 M 2.3.4 M



log. 6

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92. Although (33) gives the value of M in terms of a, yet the series in the denominator is not convergent when a is greater than 2, as will be seen by substituting 3, 4, &c. for a. It is rendered convergent by the following simple transformation, given by LAGRANGE. The base of the system being a, the logarithm of any root of a,

1 as the inth root,

is by the definition of logarithms, (Art. 58.)


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But since (34) expresses the value of the logarithm of any number, we may employ it to express that of the logarithm of am by substituting b=am. We thus obtain

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m[api—1—} (a*—1)*+}(—1–&c.] A=m[e-1-(2-1) +}(0?–1) – &c.] (13)


In this formula the series may always be rendered convergent. For


m being indefinite, we may assume it so great that am shall be less


than 2; then am—1 will be less than 1, and its powers will consequently decrease. We see also that the greater m is assumed the more rapidly will the series converge.

93. Let us apply this formula to finding the modulus of the common system in which a=10. Assume m=32; then am

am =107'7. The 32nd root of 10 may be found by five successive extractions of the square root, since 32=25. We shall find*

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* This operation will be shortened by employing the methods explained in Chapter IV, Arts. 50, 51, 52.

1 Therefore am—1=.07460,78283,2, and the calculation of the series

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The value of Mto sixty-one places is .43429,44819,03251,82765, 11289,18916,60508,22943,97005,80366,65661,14453,8.

94. It is necessary in calculations like the above, to employ throughout the process at least two more decimals than are retained in the result, so that the error occasioned by the neglected figures may not affect the last figure retained. We here employ eleven places of decimals in the calculation, but retain only nine in the result, which may therefore be relied upon as correct in the last place.

The above is the direct method of finding the modulus from the base, but it may also be obtained with the aid of naperian logarithms. (See Art. 99.)



95. This is effected at once by (41), for if b= the base of the system =a, then log. b=l, and this formula becomes

1 1



2.3.4(M)*+ &c.


In constructing a system of logarithms, we may either assume the base and then determine the modulus by Art. 92; or we may assume the modulus and then determine the base by (44). In the common system, or Briggs', the base is assumed =10; in the naperian system the modulus is assumed =1.


96. In the system of Napier, the inventor of logarithms, the modulus is assumed =1. If therefore we make M=1, in (44,) the series will give the value of Napier's base, which is generally represented by e.

We have then

e =

1 =1+1+



1 1

1 + +

+ &c. 2.3 2.3.4


The exponents of the powers of this base are naperian logarithms. They are also called hyperbolic logarithms, from the use made of them in the quadrature of the equilateral hyperbola. Although the common system has superseded the use of tables of naperian logarithms in nearly every practical operation, yet the latter possess some important advantages in theoretical investigations, and particularly in the Integral Calculus; advantages resulting principally from the simplicity of the logarithmic formula, from which the factor M is made to disappear by being assumed =1.

The value of e is found approximately by taking a number of terms of the series. The calculation is made very simple by remarking that if any term be divided by the number indicating its place in the series, the result is the next succeeding term; thus, the first term divided by 1 gives the second term 1; this divided by 2 gives the third term , &c. The calculation will therefore be as

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follows, where for convenience we designate the successive terms by A, B, C, &c.

1+1 = 2.
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.16666,66666,666 67 (B)

4166,66666,666 67 (C)
{C 833,33333,333 33 (D)
¿D 138,88888,888 89 (E)

19,84126,984 13 (F)
2,48015,873 02 (G)

27557,319 22 (H)
To H

2755,731 92 (I)
250,521 08 (K)
20,876 76 (L)
1,605 90 (M)
114 71 (N)

7 65 (0)

48 (P)

3 e= Napier's base = 2.71828,18284,590 46

The value of e to sixty-one places is 2.71828,18284,59045, 23536,02874,71352,66249,77572,46928,08355,51550,58417,2.

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97. If x represents the logarithm of b in the system whose modulus is M, and a the logarithm of b in the system whose modulus is M', we have by (34)

x=M (6-1-}(6—1)+ }(6-1)3— &c.) (46)

X'=M'(6—1—1(6-1)+ }(6-1)3— &c.) (47) Dividing (46) by (47), we have

X : X' :: M:M'.

(48) That is, the logarithms of the same number in different systems are to each other as the moduli of those systems.

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