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Another definition of logarithms is, that the logarithm of any number is the index of that power of some other number, which is equal to the given number. So if there be Nr",then n is log. of N; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the different systems of logarithms.

When n is 0, then N is =1, whatever the value of r is; which shows, that the logarithm of 1 is always O in every system of logarithms.

When n is=1, then N is =r; so that the radix r is always that number, whose logarithm is 1 in every system.

When the radix r is =2 ̊718281828459, &c. the indices n are the hyperbolic or Napier's logarithm of the numbers N; so that n is always the hyperbolic logarithm of the number Nor 2'718, &c.".

But when the radix r is =10, then the index n becomes the common or Briggs" logarithm of the number N; so that the common logarithm of any number 10 or N is n the index of that power of 10, which is equal to the said number. Thus, 100, being the second power of 10, will have 2 for its logarithm; and 1000, being the third power of 10, will have 13 for its logarithm; hence also, if 50 be =101 69897, then is 1'69897 the common logarithm of 50. And in general the following decuple series of terms,

-3

viz. 10*, 103, 103, 10', 10°, 1671, 102, 10-3, 10, or 10000, 1000, 100, 10, 1, 1, 01, 001, 0001, have 4, 3,2, 1, 0, −1, −2, —3, -4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as beforementioned.

PROBLEM,

To compute the logarithm to any of the natural numbers, 1, 2, 3, 4, 5, &c.

RULE.

Let b be the number, whose logarithm is required to be found; and a the number next less than b, so that ba=1,

the logarithm of a being known; and let s denote the sum of the two numbers a+b. Then

1. Divide the constant decimal 8685889638, &c. by s, and reserve the quotient; divide the reserved quotient by the square of s, and reserve this quotient; divide this last quotient also by the square of s, and again reserve the quotient; and thus proceed, continually dividing the last quotient by the square of s, as long as division can be made.

2. Then write these quotients orderly under one another, the first uppermost, and divide them respectively by the odd numbers, 1, 3, 5, 7, 9, &c. as long as division can be made; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, and so on.

3. Add all these last quotients together, and the sum will be the logarithm of ba; therefore to this logarithm add also the given logarithm of the said next less number a, so will the last sum be the logarithm of the number b proposed. That is, log. of b is log. a + 2 × : 1+ 12

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+. + + &c. where n denotes the constant given decimal 8685889638, &c.

EXAMPLES.

EXAMPLE 1. Let it be required to find the logarithm of the number 2.

Here the given number bis 2, and the next less number a is 1, whose logarithm is 0; also the sum 2+1=3=s, and its square s2=9. Then the operation will be as follows.

3)'868588964 1) 289529654(*289529654
3) 32169962( 10723321
3574440(

9)*289529654

9) 32169962

-5)

714888

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EXAMPLE 2. To compute the logarithm of the number 3. Here b-3, the next less number a=2, and the sum a+b= 5s, whose squares is 25, to divide by which, always multiply by '04. Then the operation is as follows.

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Then, because the sum of the logarithms of numbers gives the logarithm of their product, and the difference of the logarithms gives the logarithm of the quotient of the numbers, from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many loga-, rithms, as in the following examples.

EXAMPLE 3.

Because 2 × 2=4, therefore
To logarithm 2 301029995
Add logarithm 2 301029995

Sum is logarithm 4 602059991.

EXAMPLE 4.

Because 2×3=6, therefore
To logarithm 2 301029995
Add logarithm 3 477121255

Sum is logarithm 6 778151250.

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And thus, computing by this general rule, the logarithms to the other prime numbers 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the table.*

*Many other ingenious methods of finding the logarithms of numbers, and peculiar artifices for constructing an entire table of them, may be seen in Dr. HUTTON'S Introduction to his Tables, and Baron MASERES' Scriptores Logarithmici.

DESCRIPTION AND USE OF THE TABLE
OF LOGARITHMS.

Integral numbers are supposed to form a geometrical series, increasing from unity toward the left; but decimals are supposed to form a like series, decreasing from unity toward the right, and the indices of their logarithms are negative. Thus, +1 is the logarithm of 10, but-1 is the logarithm of, or 1; and + 2 is the logarithm of 100, but -2 is the logarithm of, or '01; and so on.

Hence it appears in general, that all numbers, which consist of the same figures, whether they be integral, or fractional, or mixed, will have the decimal parts of their logarithms the same, differing only in the index, which will be more or less, and positive or negative, according to the place of the first figure of the number. Thus, the logarithm of 2651 being 3'4234097, the logarithm of, or to, or too, &c. part

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Hence it appears, that the index, or characteristic, of any logarithm is always less by 1 than the number of integral figures, which the natural number consists of; or it is equal to the distance of the first or left hand figure from the place of units, or first place of integers, whether on the left, or on the right of it; and this index is constantly to be placed on the left of the decimal part of the logarithm.

When there are integers in the given number, the index is always affirmative; but when there are no integers, the index is negative, and it is to be marked by a short line drawn before, or above it. Thus, a number having 1, 2, 3, 4, 5, &c. integral places, the index of its logarithm is Q, 1, 2, 3, 4, &c. or 1 less than the number of those places.

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