EXPONENTIAL EQUATIONS

AND

LOGARITHMS.

SECTION I.

EXPONENTIAL EQUATIONS,

1. An Exponential Equation is one in which the unknown quantity occurs as an exponent.

2. Problem. To solve the exponential equation

bx

m.

Solution. This equation is readily solved by means of continued fractions, as explained in Alg. art. 229.

258

ΕΧΡΟΝΕΝΤIAL EQUATIONS.

Solution of Exponential Equations.

[$1.

the greatest integer contained in x must be 4. Substituting then

which being raised to the power denoted by x', is

By raising

to different powers, the greatest integer con

tained in x' is found to be 5. Substituting then

from which the greatest integer contained in x" is found to

be 4; and in the same way we might continue the pro

cess..

The approximate values of a are then

4, 45, 421, = 4,19, &c.

1

ΕΧΡΟΝΕΝΤIAL EQUATIONS.

Solution of Exponential Equations.

2. Find an approximate value for 2, in the equation

3 = 15.

259

Ans. z = 2,46.

3. Find an approximate value for x, in the equation

10x = 3.

- Ans. x = 0,477.

4. Find an approximate value for z, in the equation

(1)=.

Ans. x = 0,53.

3. Corollary. Whenever the values of p and m are both larger or both smaller than unity, the value of x is positive. But when one of them is larger than unity while the other is smaller, the value of a must be negative; for the positive power of a quantity larger than unity must be larger than unity, and the positive power of a quantity smaller than unity is smaller than unity; whereas the negative power, being the reciprocal of the corresponding positive power, must be greater than unity, when the positive power is less than unity, and the reverse.

Hence to solve the equation

b = m,

in which one of the quantities, b and 'm, is greater than unity, while the other is smaller than unity, make

which may be solved as in the preceding article.

is called the logarithm of m; and since, by the preceding section, this root can be found for any value which m may have, it follows that every member has a logarithm. The logarithm of a number is usually denoted by log. before it, or simply by the letter l.

5. But the value of the logarithm varies with the value of b, and therefore the value of b, which is called the base of the system of logarithms, is of great importance; and the logarithm of a number may be defined as the exponent of the power to which the base of the system must be raised in order to produce this number.

Logarithm of Product and of Power.

6. Corollary. When the base is less than unity, it follows, from art. 3, that the logarithms of all numbers greater than unity are negative, while those of all numbers less than unity are positive.

But when, as is almost always the case, the base is greater than unity, the logarithms of all numbers greater than unity are positive, while those of all numbers less than unity are negative.

7. Corollary. Since

6° = 1,

it follows that the logarithm of unity is zero in all systems.

8. Theorem. The sums of the logarithms of several numbers is the logarithm of their continued product.

Demonstration. Let the numbers be m, m', m", &c., and let b be the base of the system; we have then

log. m m' m" &c. = log. m + log. m' + log. m" + &c.

9. Corollary. If the number of the factors, m, m', &c. is n, and if they are all equal to each other, we have

or

log.mmm &c. =log. m+log. m + log. m + &c.

log. m2 = n log. m;