Adopting this method of writing the logarithms, we see that the logarithm of a decimal fraction may be found from the tables, by uniting to the logarithm of its numerator, regarded as a whole number, a negative characteristic greater by unity than the number of ciphers between the decimal point and the first significant figure. To demonstrate this in a general manner, let a denote the numerator of a decimal fraction, and b its denominator. From the nature of decimals, we shall have b = (10)m, in which m will denote the number of ciphers in the denominator. Hence a 웅 = log. (10)m) log. 어 = log. a - m log. 10 = log. a m. Or, in other words, the logarithm of a whole number will become the logarithm of a corresponding decimal, by adding to it a negative characteristic containing as many units as there are ciphers in the denominator of the decimal fraction. Hence, the table of logarithms whose base is 10, will give the logarithms of all decimals, as well as of the integral numbers. Number corresponding, 4 0.603060. 4. What is the 4th root of 81? 5. What is the 5th root of 32? Ans. 3. Ans. 2. 7. Log. (abc) 6. Log. (a.b.c.d....) = log. a + log. b + log.c.... = log. a+log.blog.c-log.d-log.e. 8. Log. (am.br.cp....) = m log. a + n log. b + p log. c +.... and suppose a to be the base of a system of logarithms. Then, that is, whatever be the base of the system, its logarithm taken in that system, is equal to 1, and the logarithm of 1 is equal to 0. 264. Let us suppose, in the equation If now, N diminishes, & will increase, and when N becomes 0, we have but no finite power of a is infinite, hence x = ∞ : and therefore, the logarithm of 0 in a system of which the base is greater than unity, is an infinite number and negative. 265. Again, take the equation a = N, and suppose the base a <1. Then making, as before, N = 1, we have ao 1. If we make N less than 1, we shall have a = N < 1. Now, if we diminish N, x will increase; for, since a 1, its powers will diminish as the exponent x increases, and when N= 0, x must be infinite, for no finite power of a fraction can be 0. Hence, the logarithm of 0 in a system of which the base is less than unity, is an infinite number and positive. Logarithmic and Exponential Series. 266. The method of resolving the equation a = b, explained in Art. 255, gives an idea of the construction of logarithmic tables; but this method is laborious when it is necessary to approximate very near the value of x. Analysts have discovered much more expeditious methods for constructing new tables, or for verifying those already calculated. These methods consist in the development of logarithms into series. Taking again the equation y, it is proposed to develop the logarithm of y into a series involving the powers of y, and co-efficients independent of y. It is evident, that the same number y will have a different logarithm in different systems, that is, for different values of the base a; hence, the log. y, will depend for its value, 1st, on the value of y; and 2dly, on a, the base of the system of logarithms. Hence, the development must contain y, or some quantity dependent on it, and some quantity dependent on the base a. To find the form of this development, we will assume log. y = A + By + Cy2 + Dy3 +, &c., in which A, B, C, &c., are independent of y, and dependent on the base a. Now, if we make y = 0, the log. y becomes infinite, and is either negative or positive, according as the base a is greater or less than unity (Arts. 264 & 265). But the second member under this supposition, reduces to A, a finite number: hence, the development cannot be made under that form Again, assume log. y = Ay + By2 + Cy3 + Dy+ +, &c. If we make y = 0, we have log. y = ± ∞, that is, ± ∞ = 0, which is absurd, and hence the development cannot be made under the last form. Hence we conclude that, the logarithm of a number cannot be developed in the powers of that number. Let us place, in the first member, 1 + y for y, and we have log. (1 + y) = Ay + By2 + Cy3 + Dy2 + &c. (1), making y = 0, the equation is reduced to log. 1 = 0, which does not present any absurdity. .. In order to determine the co-efficients A, B, C, we shall follow the process of Art. 243. Since equation (1) is true for any value of y, it will be true if we substitute z for y, and we may write log. (1 + z) = Az + Bz2 + Cz3 + Dz4 + Subtracting equation (2) from (1), we obtain (2). log. (1+y)-log.(1+z)=A(y−z)+B(y2-z2)+C(y3-23) +.. (3). The second member of this equation is divisible by y - z. Let us see, if we can by any artifice, put the first under such a form that it shall also be divisible by y log. (1 + y) - log. (1 + z) 1 2. We have, = log. (+) (1 = log. (1+). can be regarded as a single number u, we can develop log. (1+u), or log. (1 + ner as log. (1 + y), which gives y-z y -2 -), 2 in the same man 3 - log. (1+1)=A+B (1)2 +c()*+.. Substituting this development for 1+2 log. (1 + y) - log. (1 + z), y in the equation (3), and dividing both members by y - z, it be comes A 1 = A + B (y + z) + C (y2 + yz + z2) + ... ... Since this equation, like the preceding, is true for all values of y and z, make y = z, and there will result A 1+ y = A + 2By+3Cy2 + 4Dy3 + 5Ey2 + whence, by making the terms entire, and transposing, Placing the co-efficients of the different powers of y equal to zero, we obtain the series of equations A-A = 0, 2B + A = 0, 3C + 2B = 0, 4D + 3C = 0 ...; The law of the series is evident; the co-efficient of the nth term A is equal to -, according as n is even or odd: hence, we ob n Hence, although the logarithm of a number cannot be developed in the powers of that number, yet it may be developed in the powers of a number less by unity. By the above method of development, the co-efficients B, C, D, E, &c., have all been determined in functions of A; but A remains entirely undetermined. This indeed should be so, since A depends for its value on the base of the system, to which any value may be assigned. Denote by a that part of the second member of equation (4) which involves y, and suppose a to be the base of the system in which the log. (1 + y) is taken, and we have aAx/ = 1 + y, or Ax' = log. (1 + y). But the log. (1 + y) depends for its value on two things: viz., on the number of units in y, and on the base of the system in which the logarithm is taken. The series denoted by x' is ex |