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 bxm. Solution. This equation is readily solved by means of continued fractions, as explained in Alg. art. 324. Solution of Exponential Equations. the greatest integer contained in x must be 4. Substituting which being raised to the power denoted by x', is By raising to different powers, the greatest integer contained in ' is found to be 5. Substituting then 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, Solution of Exponential Equations. 2. Find an approximate value for x, in the equation 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 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 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 number has a logarithm. The logarithm of a number is usually denoted by log. before it, or simply by the letter l. 7. 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. |