either be set aside and added to the fractional part after its value shall have been found, or we may place 1 under it for a denominator and treat it as an approximating fraction.

Of Exponential Quantities.

Resolution of the Equation a=b.

255. The object of this question is, to find the exponent of the power to which it is necessary to raise a given number a, in order to produce another given number b.

Suppose it is required to resolve the equation 2*=64. By rais ing 2 to its different powers, we find that 2o=64; hence x=6 will satisfy the conditions of the equation.

Again, let there be the equation 3a=243. The solution is x=5. In fact, so long as the second member b is a perfect power of the given number a, x will be an entire number which may be obtained by raising a to its successive powers, commencing at the first.

Suppose it were required to resolve the equation 2-6. By making x=2, and x=3, we find 22=4 and 23=8: from which we perceive that has a value comprised between 2 and 3.

or

x

1 x'

Suppose then, that x=2+ in which case x'>1.

Substituting this value in the proposed equation, it becomes,

(2)=2, by changing the members, and raising both to the

=2

By substituting this value in the equation (-)"=

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

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 I will be comprised. Making

so on.

1

x can be determined in the same manner as xv, and

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 value of the fractional part to within

gree 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 a*=y,

and y to be replaced by all possible positive numbers, it is plain that ≈ 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.

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

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 dimin: ishes as the value of x increases negatively.

1

Suppose a<1 or equal to the proper fraction —.

a

y=(-/-)'=

1

&c.

a''

x=0, −1,

-3, -4,

=(1/2)=1,

a', a', a'3, a', ... &c.

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.

2d,

a=1, whence log 1=0.

The word logarithm is commonly denoted by the first three letters log, or simply by the first letter 1.

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

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