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As the difference of the errors, to the dif-
ference of the assumed numbers;

So is the least error, to the correction required
in the corresponding assumed number.

Ex. 1. Find the value of x in the equation *= 256. Taking the logarithms of both sides (log. x) xx=log. 256. Let x be supposed equal to 3.5, or 3.6.

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Then, 0.09844 : 0.1 :: 0.40556 0.4119, the correction. This added to 3.6, the, second assumed number, makes the value of x=4.0119.

To correct this farther, suppose x=4.011, or 4.012.

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Subtracting this correction from the first assumed number 4.011, we have the value of x=4, which satisfies the conditions of the proposed equation; for 4+=256.

2. Reduce the equation 4x100x3.

3. Reduce the equation x=9x,

Ans. x=5.

64. The exponent of a power may be itself a power, as in the equation

aTM3=b;

where x is the exponent of the power m3, which is the exponent of the power am”.

Ex. 4. Find the value of x, in the equation 93 -1000.

3x (log. 9)=log. 1000. Therefore, 35=

Then, as 3=3.14. ≈ (log. 3)=log. 3.14.

Therefore, x=

log. 3.14

log. 3

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log. 1000
log. 9

= 3.14.

In cases like this, where the factors, divisors, &c., are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus, the value of the fraction !!!!! is most easily found, by subtracting the logarithm of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator.

ba+d

5. Find the value of x, in the equation --=m.

Ans. x=

6

с

log. (cmd)- log. b. log. a.

SECTION IV.

DIFFERENT SYSTEMS OF LOGARITHMS, AND COMPUTATION OF THE TABLES.

65. For the common purposes of numerical computation, Briggs' system of logarithms has a decided advantage over every other. But the theory of logarithms is an important instrument of investigation, in the higher departments of mathematical science. In its numerous applications, there is frequent occasion to compare the common system with others; especially with that which was adopted by the celebrated inventor of logarithms, Lord Napier. In conducting these investigations, it is often expedient to express the logarithm of a number, in the form of a series.

If a*= N, then x is the logarithm of N. (Art. 2.)

To find the value of x, in a series, let the quantities a and N be put into the form of a binomial, by making a=1+b, and N=1+n. Then (1+6)=1+n, and extracting the root y of both sides, we have

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1

Ꮖ y y

As these expressions will be the same, whatever be the value of y, let y be taken indefinitely great; then and being indefinitely small, in comparison with the numbers -1, -2, &c., with which they are connected, may be cancelled from the factors

-2

(~—-—1) (~—-—2), &c. (———1), ((——2), &c.

(Alg. 456.) leaving 1+)+(G) (4), &c.

1

y

1

3

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Rejecting 1 from each side of the equation, multiplying by y, (Alg. 159.) and dividing by the compound factor into which x is multiplied, we have

++
x=Log. N=2-n3+}n3 —\n•+ &c.
b -162 +163
bb2 + b3 — b1+ &c.

Or, as n N-1, and b-a-1,

Log. N

=

=

A

(N −1) —¦(N —1)3 +}(N −1)3 —}(N−1)1+ &c. ( a −1)—(a —1)2 + ( a −1)3 —(a —1)1+ &c. Which is a general expression, for the logarithm of any number N, in any system in which the base is a. The numerator is expressed in terms of N only; and the denominator in terms of a only: So that, whatever be the number, the denominator will remain the same, unless the base is changed. The reciprocal of this constant denominator, viz.

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-

1

1) — (a−1)3 +1(a—1)3—(a-1)+ &c. is called the Modulus of the system of which a is the base. If this be denoted by M, then

Log. N=M×((N−1)—}(N −1)2+}(N−1)3 —‡(N −1)• + &c.)

66. The foundation of Napier's system of Logarithms is laid, by making the modulus equal to unity. From this condition the base is determined. Taking the equation above marked A. and making the denominator equal to 1, we have x = n = {n2 + { n3 — { n2 + {n3 — &c. By reverting this equation

*

x2 X3 X4

n = x+ + + +:

X5

+ &c. 2 2.3 2.3.4 2.3.4.5

Or, as by the notation, n +1=N=a*,

x2 X3 X4

a2=1+x+ +

X5

+
22.3 2.3.4 2.3.4.5+ &c.

If then x be taken equal to 1, we have

1 1 1

1

a=1+1+2+2.3 2.3.4 2.3.4.5
+2.3.4.5+ &c.

Adding the first fifteen terms, we have

2.7182818284

Which is the base of Napier's system, correct to ten places

of decimals.

*See note D.

Napier's logarithms are also called hyperbolic logarithms, from certain relations which they have to the spaces between the asymptotes and the curve of an hyperbola; although these relations are not, in fact, peculiar to Napier's system.

67. The logarithms of different systems are compared with each other, by means of the modulus. As in the series

(N−1)—(N—1)2 +¦(N −1)3—1(N −1)1+ &c.
(a-1)-(a-1)3 +} (a—1)3—4 (a-1)++ &c.

which expresses the logarithm of N, the denominator only is affected by a change of the base a; and as the value of fractions, whose numerators are given, are reciprocally as their denominators: (Alg. 360. cor. 2.)

The logarithm of a given number, in one system,

Is to the logarithm of the same number in another system; As the modulus of one system,

To the modulus of the other.

So that, if the modulus of each of the systems be given, and the logarithm of any number be calculated in one of the systems; the logarithm of the same number in the other system may be calculated by a simple proportion. Thus, if M be the modulus in Briggs' system, and M' the modulus in Napier's; the logarithm of a number in the former, and l' the logarithm of the same number in the latter; then,

M: M'::: l',

Or, as M'=1,
M: 1::/: l'.

Therefore, l-l'x M; that is, the common logarithm of a number, is equal to Napier's logarithm of the same, multiplied into the modulus of the common system.

To find this modulus, let a be the base of Briggs' system, and e the base of Napier's; and let l.a denote the common logarithm of a, and la denote Napier's logarithm of a.

Then, M: 1 :: l.a: l'.a.

l.a

Therefore, M=

Ζ ́.α

But in the common system, a=10, and l.a=1.

1 7.10'

So that, M= that is the modulus of Briggs' system, is

equal to 1 divided by Napier's logarithm of 10.

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