The logarithms of the intermediate numbers, from what has been already said, will consist of an integer and of a decimal part; the decimal part alone is registered in the tables, and is called the mantissa of the logarithm; the integer part, which is called the characteristic, is not entered in the tables, but is found by the following rule: 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 is found, by referring to the tables, to be .9289077; therefore

log 849 2.9289077;

=

and therefore from the definition of a logarithm,

849

=

102.9289077.

The logarithms of numbers less than unity are negative, as will be seen from the following table:

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

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, I, 2, 3, &c. The reason of the rule just given may be seen from inspecting the preceding table of values of negative powers of 10, or perhaps more clearly by discussing a particular case; for example, the one already proposed, viz.:

If we divide each side of this equation successively by 10, which is effected on the left-hand side by removing the decimal point one place to the left; on the right-hand side, by subtracting unity from the index of the power, we obtain,

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

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

PROPERTIES OF LOGARITHMS.

2. We shall now proceed to demonstrate the properties of loga. rithms, and lay down the rules for using logarithmic tables in numerical computations.

Let n = log N, and m = log M; then

N = 10",

M = 10m.

If we multiply these,

Nx M=10n+m;

therefore,

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

PROPOSITION I.

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

If we divide the former of these equations by the latter

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

If we raise each side of the equation

The logarithm of the pth power of a number is equal to p times the logarithm of the number.

If we take the pth root of each side of the equation

PROPOSITION IV.

The logarithm of the pth root of a number is equal to the pth part of the logarithm of the number.

Upon these propositions depend the rules for applying logarithmic tables to calculations. Previously to giving any examples of such calculations, we shall describe the method of using the tables.

USE OF LOGARITHMIC TABLES.

3. In using the tables two questions present themselves, viz. :Being given a number, it is required to find its logarithm. Being given the logarithm of a number, it is required to find the number.

The logarithms of numbers are found by the following rules.
If the number consist of less than six figures, we proceed by

RULE I.

The first four figures are found in the vertical column on the left, marked N; and the fifth in the horizontal column, at the top of the Under this, and opposite the four figures, will be found the The characteristic is found by the rules in section 1.

page. mantissa.

If the number of digits be less than five, conceive ciphers annexed,

so as to make up five, and proceed as above.

5. Find the logarithm of 127.

9.

6.

832.

632.
5421.

Ans. 2.1038037.

Ans. 1.3222193.

Ans. 0.9542429.

Ans. 2.8007171.

Ans. 3.7340794.

N. B.-It may be observed that the mantissæ are arranged in sets, which have the first three figures common. In the last line of the set we sometimes find a bar printed over the first of the last four figures; this signifies that the first three figures of this and the remaining mantissæ of the line are to be taken from the next set. For example, if we look for the logarithm of 65766, we find that the last

four figures are Ō014; the bar signifies that the first three figures are 818, and not 817; the mantissa is, therefore, .8180014. For the same reason, the mantissa corresponding to 65769 is 0.8180212.

If the number consist of more than six figures, we use

RULE II.

1°. Cut off the first five figures, and consider the rest as a decimal. 2°. Find the mantissa corresponding to the first five figures.

3°. Multiply the tabular difference by the decimal cut off.

4. Add the integer part of this product to the last figures of the mantissa just found.

The result is the mantissa of the required logarithm. The characteristic is found by the rules in section 1.

In the following examples we find the mantissa which corresponds to the significant figures, and prefix the negative characteristics ac