Hence, to find the cube root of 21.75 by the formula, we have a2.758924, n.75,

d1.043115, d2.001287, ds = +.000091, etc. These values substituted in the formula, give

If it were required to find the cube root of any number between 22 and 23, we might put n equal to the excess of the number above 21, and employ the same values for d1, da, dg, etc., as before. But greater accuracy will be attained by making 22 the first term of the series, and employing the corresponding differences; in which case n will be a proper fraction.

398. On three successive days, the angular distances of the sun from the moon, as seen from the earth, were as follows:

In the data here given, the interval of time is 12 hours. Hence, to find the distance of the sun from the moon at intermediate times, n must always be some fractional part of 12. Thus, for the distance at 3 o'clock P. M. of the first day we have n=1, and a = 66° 6′ 38′′; for the distance at 6 o'clock A. M. of the second day, n, and a = 72° 24' 5". For the distance at 3 o'clock P. M. of the second day, n=4, and a 78° 34' 48".

=

EXAMPLES FOR PRACTICE.

Find by interpolation the distance of the sun from the moon,

LOGARITHMS.

399. The Logarithm of a number is the exponent of the power to which a certain other number, called the base, must be raised, in order to produce the given number. Thus, in the expression, a=b,

the exponent, x, is the logarithm of b to the base a.

An equation in this form is called an exponential equation.

If in this equation we suppose a to be constant, while b is made equal to every possible positive number in succession, the corresponding values of x will constitute a system of logarithms; hence,

400. A System of Logarithms consists of the logarithms of all possible positive numbers, according to a given base.

Any positive number except unity may be made the base of a system of logarithms. For, by giving to x suitable values, the equation απ b

will be true for all possible positive values of b, provided a is any positive number except 1. Hence,

There may be an indefinite number of systems of logarithms.

401. If in the equation ab, we suppose b to represent a perfect power of a, then x will be some integer; but if b is not a perfect power of a, then x will be some fraction. Hence,

A logarithm may consist of an integral and a fractional part. 402. The Index or Characteristic of a logarithm is the integral part; and

403. The Mantissa is the fractional part of a logarithm. For illustration, let 5 be the base of a system; then we have 52.25=5= √59 37.384.

=

Thus, the logarithm of 37.384 to the base 5, is 2.25; the index of this logarithm is 2, and the mantissa .25.

PROPERTIES OF LOGARITHMS.

404. There are certain properties of logarithms, which are common to all systems. To investigate these general properties,

let a denote the base of the system; also, designate the logarithm

of a quantity by log., written before the quantity.

1. In any system, the logarithm of unity is 0.

2. In any system, the logarithm of the base is unity.
For, let
a2 = a; then x= log. a.

But by 88, if a® = a, then x = 1, or log. a = 1.

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

But by multiplication,

mn = a2+2;

therefore, log. mn = x + z = log. m + log. n.

4. The logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor.

For, let

then

By division,

m =

a*,

log. m,

n = a2;
2= log. n.

m

X=

5. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. For, let

By involution,

therefore,

m = a; then x = log. m.

6. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root.

405. The principal use of logarithms is to facilitate arithmetical computations. By means of the last four properties, we may avoid the ordinary labor of multiplication, division, involution, and evolution,-these operations being practically performed by addition and subtraction.

For this purpose, it is necessary to have a Table of Logarithms, so constructed that we may readily obtain the logarithm of any number within a certain limit, or the number corresponding to any logarithm, to a certain degree of approximation. The common tables give the logarithms of numbers from 1 to 10,000, correct to 6 decimal places.

With a table of this kind, we have the following obvious

RULES FOR COMPUTATION.

I. To multiply one number by another:-Find the logarithms of the given numbers; add these logarithms, and find the number corresponding to the sum; this number will be the required product (404, 3).

II. To divide one number by another :-Find the logarithms of the given numbers; subtract the logarithm of the divisor from that of the dividend, and find the number corresponding to the difference; this number will be the required quotient (404, 4).

III. To raise a number to any power :-Find the logarithm of the given number, and multiply it by the exponent of the required power; then find the number corresponding to this product, and it will be the required power (404, 5).

IV. To extract any root of a number :- Find the logarithm of the given number, and divide it by the index of the root; then fină the number corresponding to the quotient, and it will be the required root (404, 6).

NOTE.-From 400, we infer that negative numbers, as such, have no logarithms. But we may always employ logarithms in calculations where negative factors are involved, by disregarding signs until the absolute value of the product or quotient is obtained.