Make,

9

2;

4

x = 1 + in which ">1.

By substituting this value in the equation (2) = 2,

therefore, x" is comprised between 1 and 2.

Operating upon this exponential equation in the same manner as upon the preceding equations, we shall find two entire numbers, k and k + 1, between which av will be comprised. Making

and av can be determined in the same manner as IV, and so on. Making the necessary substitutions in the equations

we obtain the value of x under the form of a continued fraction

Hence, we find the first three approximating fractions to be

which is the true value of the fractional part of x to within

and if a greater degree of exactness is required, we must take a greater number of integral fractions.

Theory of Logarithms.

256. If we suppose a to preserve the same value in the equation ax = N,

and N become, in succession, every possible positive number, it is plain that a will undergo changes corresponding to those made in N. By the method explained in the last Article, we can determine, for each value of N, the corresponding value of x, either exactly or approximatively.

Any number, except 1, may be taken for the invariable number a; but when once chosen, it is supposed to remain the same for the formation of one entire series of numbers.

The exponent x of a, corresponding to any value of N, is called the logarithm of that number; and the invariable number a is called the base of that system of logarithms. Hence,

The logarithm of a number, is the exponent of the power to which it is necessary to raise an invariable number, called the base of the system, in order to produce the number.

The general properties of logarithms are independent of any particular base. The use that may be made of them in numerical calculations, supposes the construction of a table, containing all the numbers in one column, and the logarithms of these numbers in another, calculated from a given base. Now, in calculating this table, it is necessary, in considering the equation a** = N,

to make N pass through all possible states of value, and to determine the value of x corresponding to each of the values of N, which may be done by the method of Art. 255.

257. The base of the common system of logarithms, or as they are sometimes called, Briggs' logarithms, from their inventor, is che number 10. If we designate the logarithm of any number . by log. or l, we shall have

Hence, in the common system, the logarithm of any number between 1 and 10, is >0 and <1. The logarithm of any number between 10 and 100, is > 1 and <2; the logarithm of any number between 100 and 1000, is > 2 and < 3; and so on. Hence, the logarithm of any number expressed by two figures. and which is not a perfect power of the base of the system, will be equal to a whole number plus an approximating fraction, the approximate value of which fraction is generally expressed decimally.

The integral part of a logarithm, is called the index or characteristic of the logarithm.

By examining the several powers of 10, we see, that if a number is expressed by a single figure, the characteristic of its logarithm will be 0; if it is expressed by two figures, the characteristic of its logarithm will be 1; if it is expressed by three figures, the characteristic will be 2; and if it is expressed by n places of figures, the characteristic will be n . 1 units.

The following table shows the logarithms of the numbers, from 1 to 100.

The characteristic being always one less than the number of places of figures in the number, is not written down in the table of logarithms for numbers which exceed 100. Thus, in searching for the logarithm of 2970, we should find in the table oppc site 2970, the decimal part .472756. But since the number is expressed by four figures, the characteristic of the logarithm is 3.

Hence,

log. 2970

3.472756,

and by the definition of a logarithm, the equation

a2 = N, gives

103.472756 = 2970.

Multiplication and Division by Logarithms.

258. Let a be the base of a system of logarithms, and suppose the table to be calculated. Let it be required to multiply together a series of numbers by means of their logarithms. Denote the numbers by N, N', N', N'", &c., and their corresponding logarithms by x, x, x, x", &c. Then, by definition (Art. 257), we have

a2 = N', a2' = N', ax" = N"', a*'"' = N'"' . . . &c. Multiplying these equations together, member by member, and applying the rule for the exponents, we have

But since a is the base of the system, we have

x + x + x + x'"!

=

log. (N, N', N', N''' . . );

that is, the sum of the logarithms of any number of factors, is equal to the logarithm of the product of those factors.

259. Suppose it were required to divide one number by another. Let N and N' denote the numbers, and x and x their logarithms We have the equations

at = N and

ax' = N';

that is, the difference between the logarithm of the dividend and the logarithm of the divisor, is equal to the logarithm of the quotient.