and then the value of x may be found from the final 363 =m, whence, log m = log 3 + 3 log b-2 log a; (a.) a2 64 a3 =n, whence, log n = 4 log b - 3 log a; 462c ar, log log 4+2 log b + log e 2 log a; (c.) a2 we find, r = c log x = log a + log (2a — m + n) — log (a - 36 +r). m, n, and r, may be found from Equations (a), (b), and (c), and then x may be computed from the final equation. 8. Transform the equation, x = Ans. log x = loga + log b + † log c. The General Principles serve also to solve exponential equations, that is, equations in which the unknown quantities enter an exponent. Solve the following equations : 9. 7 13. Taking the logarithms of both. members, we have, First find the value of d, then that of x. 14. a*by = c) my = пх Taking the logarithms of both members of the first equation, we have, x log ay log blog c. Combining this with the second of the given equations, Taking the logarithms of both members of the first equation, remembering that the logarithm of 2000 is equal to the logarithm of 1000 x 2, equal to 3+ log 2, we have, x log 2+ y log 3 3+ log 2; = which, combined with the second of the given equa tions, gives, 186. There are certain general properties of loga. rithms that may be discovered by a discussion of the exponential equation, ax = n (1.) In this equation, the arbitrary quantities are a and n. = First. If we make n 1, the corresponding value of x will be 0, whatever may be the value of a, since ao = 1, (Art. 31). Hence, The logarithm of 1, in any system whatever, is equal to 0. Second. If we make n = a, the corresponding value of a will be 1, whatever may be the value of a. Hence, The logarithm of the base of any system, taken in that system, is 1. Third. If we suppose a > 1, say 10, for example, we shall have, 10 n. When n = 1, the value of x, or the logarithm of 1, is 0; when n = ∞, the value of x, or the logarithm , iso. The logarithms of all numbers between of 1 and ∞, lie between 0 and ∞, that is, they are positive. 1 When n is less than 1, x must be negative, giving 10* = n; when n = 0, x will be infinite, in the last given equation, that is, the logarithm of 0 is equal In any system whose base is greater than 1, the logarithms of all numbers greater than 1, are positive; the logarithms of all numbers less than 1, are negative; the logarithm of oo, is+co, and the logarithm of 0, is∞. 1 9 10 Fourth. If we suppose a < 1, say for example, we shall have, 1 In this case, the positive values of a correspond to the negative values of x in the preceding case; and the negative values of x, to the positive values, in the preceding case. Hence, In any system whose base is less than 1, the logarithms of all numbers greater than 1, are negative; the logarithms of all numbers less than 1, are positive; the logarithm of ∞, is - ∞, and the logarithm of 0, is + co. со Fifth. Since, for every value of a between and, that is, for every real value of x, the values of n lie between 0 and +∞, whether a is greater or less than 1, it follows that there are no real values of x, which, substituted in the equation, a = n, will make n negative. Hence, There are no real logarithms corresponding to negative numbers. Although there are no logarithms of negative numbers, we may multiply negative numbers by means of their logarithms. We must regard the numbers as positive; and, having applied the rules, we must give the proper sign to the result, according to the rule of signs. Thus, to multiply 27 by 435, we find the pro duct of 27 and 435, and give to it the minus sign |