Imagens das páginas
PDF
ePub

EXPONENTIAL EQUATIONS

AND

LOGARITHMS.

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. 324.

[blocks in formation]

Solution of Exponential Equations.

the greatest integer contained in x must be 4. Substituting

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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

[blocks in formation]

By raising

81

to different powers, the greatest integer contained in ' is found to be 5. Substituting then

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

from which the greatest integer contained in " is found to be 4; and in the same way we might continue the process. The approximate values of x are, then,

[blocks in formation]
[ocr errors]

Solution of Exponential Equations.

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

3x = 15.

Ans. x 2.46.

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

10 = 3.

Ans. x 0.477.

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

[blocks in formation]

4. Corollary. Whenever the values of b 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 x 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

bx = m,

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

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

which may be solved as in the preceding article.

« AnteriorContinuar »