Making successively x=1, 2, 3, we find for the two last hypo

theses()

()

2 81

=

17 64 64'

1

1+ , which is <1+, and

3 729

217

=1+

1

512 512, which is >1+: therefore a" is com

Operating upon this exponential equation in the same manner as upon the preceding equations, we shall find two entire num

bers k and k+1, between which will be comprised. Making

1 =k+ αν

so on.

αν can be determined in the same manner as av, and

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

Hence we find the first three approximating fractions to be.

3×2+1

7

5×2+2-12 (Art. 252),

which is the value of the fractional part to within

gree of exactness is required, we must take a greater number of integral fractions.

EXAMPLES.

2,46 to within 0,01.
0,477
0,25

Theory of Logarithms.

256. If we suppose a to preserve the same value in the equation

a=y,

and y to be replaced by all possible positive numbers, it is plain that x will undergo changes corresponding to those made in y. Now, by the method explained in the last Article, we can determine for each value of y, the corresponding value of x, either exactly or ap proximatively.

Making in succession

there will result

First suppose a>1.

x=0,1,2,3,4,5,...&c.

y=a°=1, a, α2, α3, a, a,... &c.

hence, every value of y greater than unity, is produced by the powers of a, the exponents of which are positive numbers, entire or fractional; and the values of y increase with x.

Make now

x=0, -1, -2, -3, -4, -5, &c.

1

1 1 1

there will result y=a=1, ' ' ' '

a

...

hence, every value of y less than unity, is produced by the powers of a, of which the exponents are negative; and the value of y diminishes as the value of x increases negatively.

Suppose a<1 or equal to the proper fraction

x=0, 1, 2, 3, 4,
1 1 1 1

...

=1,

0

x=0, -1, -2, -3, -4,

...

&c.

&c.

0

α', α', α'3, a',... &c.

y=()=1,

That is, in the hypothesis a<1, all numbers are formed with

the different powers of a, in the inverse order of that in which they are formed when we suppose a>1.

Hence, every possible positive number can be formed with any constant positive number whatever, by raising it to suitable powers.

REMARK. The number a must always be different from unity, because all the powers of 1 are equal to 1.

257. By conceiving that a table has been formed, containing in one column, every entire number, and in another, the exponents of the powers to which it is necessary to raise an invariable number, to form all these numbers, an idea will be had of a table of logarithms. Hence,

The logarithm of a number, is the exponent of the power to which it is necessary to raise a certain invariable number, in order to produce the first number.

Any number, except 1, may be taken for the invariable number; but when once chosen, it must remain the same for the formation of all numbers, and it is called the base of the system of logarithms. Whatever the base of the system may be, its logarithm is unity, and the logarithm of 1 is 0.

For, let a be the base: then

1st, we have a1=a, whence log a=1.

The word logarithm is commonly denoted by the first three letters log, or simply by the first letter l.

We will now show some of the advantages of tables of logarithms in making numerical calculations.

Multiplication and Division.

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 y, y', y", y'" . &c., and their corresponding logarithms

by x, x', x", x'', &c. Then by definition (Art. 257), we have a=y, ay', a"=y", a""""=y"... &c.

Multiplying these equations together, member by member, and

applying the rule for the exponents, we have

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 y and y denote the numbers, and x and their logarithms. We have the equations

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

Consequences of these properties. A multiplication can be performed by taking the logarithms of the two factors from the tables, and adding them together; this will give the logarithm of the product. Then finding this new logarithm in the tables, and taking the number which corresponds to it, we shall obtain the required product. Therefore, by a simple addition, we find the result of a multiplication.

In like manner, when one number is to be divided by another, subtract the logarithm of the divisor from that of the dividend, then find the number corresponding to this difference; this will be the required quotient. Therefore, by a simple subtraction, we obtain the quotient of a division.