corresponding to the resulting logarithm, and it will be the root required. No practical examples can be given to illustrate the preceding rules, without a table of logarithms. A few examples of transformation are annexed, which show the methods of proceeding in the employment of logarithms. EXAMPLES. 1. Transform the equation, x = From equations (5) and (6), using common logarithms, we have, log x = log a + log b + log c abc 2. Transform the equation, x = From equations (5), (6), (7), and (8), we have, Solve the following equations: 6. 13. Taking the logarithms of both members, we have, Taking the logarithms of both members of the first equation, we have, x log a + y log b = log c. Combining this with the second of the given equations, General Properties of Logarithms. 189. There are certain general properties of logarithms that may be discovered by a discussion of the exponential equation, ax = n (1) In this equation, the arbitrary quantities are a and n. 1o. If we make n = 1, the corresponding value of x will be 0, whatever may be the value of a, since a = 1; hence, The logarithm of 1, in any system, is equal to 0. 2o. If we make na, the corresponding value of x 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. 4 3°. If we suppose a > 1, say 10, for example, we shall have, 10x = n. If n = 1, the value of x, or the logarithm of 1, is 0; if no, the value of x, or the logarithm of, is. The logarithms of all numbers between 1 and , lie between 0 and, that is, they are positive. 1 If n is less is less than 1, x must be negative, giving = n; if n = 0, x will be infinite, in the last equation, because 102 1 10 = 0, therefore, x = in the given equation, that is, the logarithm of 0 is equal to -; hence, 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, is +, and the logarithm of 0 is S. 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 + ∞. 5°. Since, for every value of x between ss 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 logarithms. We first regard the numbers as positive; and, having applied the rules, we then give the proper sign to the result, according to the rule for signs. Thus, to multiply 27 by - 435, we find the product of 27 and 435, and give it the minus sign. Logarithmic Series. 190. Let it be required to develop Log (1+ y) into a series arranged according to the ascending powers of y. If we make y = 0 in the expression Log (1 + y), it reduces to Log 1, which is equal to 0; hence, the series must be such that it will reduce to 0 when y = 0, that is, every term must contain y as a factor. We may therefore assume the series, Log (1+ y) = My + Ny2 + Py3 + Qy1 + &c. (1) in which M, N, P, &c., are constants to be determined. Since equation (1) must be true for all values of y, we may write z for y, giving the equation, Log (1 + z) = Mz + N22 + Pz3 + Qz1 + &c. (2) Subtracting (2) from (1), member from member, and remembering that Log (1+ y)-Log (1+ z) = Log we have, (1+32) = Log y = Log (1+1=3) Log (1+2—2) = M(y−2) + N(y2—z2) + P(y3—2o) +&c. (3) Every term of the second member of (3) is divis ible by y Writing z by means of formula 9, Art. 36. y 2 1+% for y in equation (1), we have, |