Solution of Exponential Equations. the greatest integer contained in a must be 4. Substituting then 3 = = which being raised to the power denoted by x', is 100\x' 81 By raising 189 to different powers, the greatest integer contained in x' is found to be 5. Substituting then 1 x' =5+ x we have 3= (1 3)*" 10000000000 100 Х 81 Hence x" 100 1.0460353 81' from which the greatest integer contained in 2'' is found to be 4; and in the same way we might continue the process. The approximate values of x are, then, 4, 43, 44, = 4:19, &c. Solution of Exponential Equations. 2. Find an approximate value for x, in the equation 3t = 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 (12)* = =* Ans. x = 0:53. 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 7* m, in which one of the quantities, b and m, is greater than unity, while the other is smaller than unity, make Y, which gives 6-Y=m, or ( ) which may be solved as in the preceding article. = mg 2. Solve by approximation the equation 2* = 5 Ans. x=-1.58. SECTION II. NATURE AND PROPERTIES OF LOGARITHMS. 6. The root of the equation = m 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. Logarithm of Product and of Power. 8. 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. 9. Corollary. Since 6 = 1, it follows, that the logarithm of unity is zero in all systems. 10. Theorem. The sums of the logarithms of several numbers is the logarithm of their continued product. Proof. Let the numbers be m, m', m'', &c., and let 6 be the base of the system ; we have then b = m', b log. m' in", &c.; the product of which is, by art. 28, blog. m + log. mi + log. mel + &c. = m m' m" &c. Hence, by art. 7, log. m m' m'' &c.= - log. m + log. m' + log. m' + &c. 11. Corollary. If the number of the factors, m, m', &c. is n, and if they are all equal to each other, we have log. mmm &c. = log. m + log. m + log. m +&c. or log. mn = n log. m; Logarithm of Root, Quotient, and Reciprocal. that is, the logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. 12. Corollary. If we substitute P = m", or m = P, in the above equation, it becomes log. p = n log. ✓ P, or that is, the logarithm of any root of a number is equal to the logarithm of the number divided by the exponent of the root. 13. Corollary. The equation log. m m' log. m + log. m', gives log. m m that is, the logarithm of one factor of a product is equal to the logarithm of the product diminished by the logarithm of the other factor ; or, in other words, The logarithm of the quotient is equal to the logarithm of the dividend, diminished by the logarithm of the divisor. log. m' - log. m; n 14. Coroltary. We have, by arts. 13 and 9, 1 log. n; that is, the logarithm of the reciprocal of a number is the negative of the logarithm of the number. |