INTRODUCTION.

1. Definition of Logarithms.-2. The Common System.-3. Properties of Loga

rithms.-4. Logarithmic Tables.-5. Multiplication by Logarithms.-6. Division by Logarithms.—7. Involution by Logarithms.-8. Évolution by Logarithms.-9. Tables of Logarithmic Sines.-10. Gauss's Logarithms.

1. Definition of Logarithms.-Let any number a be raised to the power n, and let the result be N; then

an = N

In this equation n is said to be the logarithm of the number N to the base a; and therefore

loga N

2. The Common System.- The base of the Common System,

If this number or, as it is sometimes called, Briggs' System, is 10. be raised to the powers, o, 1, 2, 3, 4, &c., we obtain the series of numbers, 1, 10, 100, 1000, 10000, &c." Thus:

100 = 1
101 = 10
102 = 100
103 = 1000

= I

= 2

&c., &c. Therefore,

log 1
log 10
log 100
log 1000 = 3

&c., &c. It is evident that for numbers intermediate to these, the powers to which 10.must be raised, must lie between the numbers of the series, 0, 1, 2, 3, 4, &c. Thus, for all numbers lying between 100 and 1000, the corresponding power of 10 being greater than 2, and less than 3, must be 2, increased by some decimal fraction; for numbers lying between 1000 and 10000, must be 3, increased by some decimal fraction; and so on.

The logarithms, therefore, of intermediate numbers consist of an integer and of a decimal part; the decimal part alone is registered in the Tables, and is called the Mantissa ; the integer part, which is called the Characteristic, is not entered, but may be found by

RULE I.

The characteristic of the logarithm of a number greater than unity is one less than the number of the digits of its integer part.

Thus, the characteristic of the logarithm of 849 is 2; for as this number lies between 100 and 1000, its logarithm must lie between 2 and 3. The mantissa, by referring to the Tables, is found to be 92891; therefore

log 849 = 2.92891 The logarithms of numbers less than unity are negative, as may be seen from the following table :

10-1= to

ido Tobo

Todoo &c.

&c. It follows from this, that the characteristics of the logarithms of all numbers less than unity are negative, and may be found by

= 0.1

IO
IO

= 0.01 = 0.001

-3

10-4

= 0.0001

RULE II.

The characteristic of the logarithm of a number less than unity, and reduced to the decimal form, is negative, and one greater than the number of ciphers following the decimal point.

A negative characteristic is denoted by writing over it the negative sign; thus, ī, 2, 3, &c. The reason of the rule may be seen from inspecting the preceding table of values of negative powers of so, or perhaps more clearly by considering a particular case ; for example,

log 849 = 2.92891 therefore

849 =

= 102.92891

If we divide each side of this equation successively by 10, we obtain,

84.9

101.92891
100.92891

8.49

from which it follows that

log 84.9 1.92891

log 8.49 = 0.92891 This corresponds with the rule given for determining the characteristic of the logarithm of a number greater than unity. If we continue the division of each side by 10, we obtain

= 101.92891

0.849
0.0849 = 102.92891
0.00849 = 103.92891

&c. &c. From which it follows that

=

log o.849 = 1.92891
log 0.0849 = 2.92891
log o'00849 = 3.92891

&c. &c. From this it appears that the negative characteristics are given by the rule in question. It also appears that the logarithms of all numbers, eonsisting of the same significant figures, have the same mantissa ; and that the characteristic depends solely on the position of the decimal point.

3. Properties of Logarithms. We shall now demonstrate four general propositions, from which the rules for using logarithmic tables in numerical computations are derived.

PROPOSITION I. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

If the numbers be N and M, let n = log N, and m= log M to any base

а,
then by the definition,

N=an
M

By multiplication,

NXM=

= antm

therefore,

log N*M = n + m = log N + log M.

PROPOSITION II.

The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers. By division.

N

The logarithm of the pth power of a number is equal to p times the logarithm of the number. If we raise to the pth power each side of the equation

N = an we obtain

NP = therefore

log NP = pn=p log N

= apn

PROPOSITION IV.

The logarithm of the pth root of a number is equal to the pth part of the logarithm of the number. If we take the pth root of each side of the equation

N= we obtain

4. Logarithmic Tables.-If the number be given, its logarithm may be found by the following rules.