log. 1. a denote the base of the system; also, designate the logarithm of a quantity by log., written before the quantity. 1.-In any system, the logarithm of unity is 0. a* = a; then x= log. a. 3.-The logarithm of the product of two numbers is equal to the sum of the logarithms of the two numbers. For, let m = a*, n=a'; then z= log. n. But by multiplication we have = log. m, mn therefore, log. mn=x+2= log. m+log. n. 4.--The logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. For, let m = a*, n = a*; then log. m, z= log. n. By division we have ; m n 5.-The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. For, let m = a*; then x = log. m. By involution we have mm = am ; therefore, log. (ma) =rx = r log. m. 6.—The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. For, let m = a*; then x = - log. m. 405. The principal use of logarithms is to facilitate arithmetical computations. By means of the last four properties, we may avoid the ordinary labor of multiplication, division, involution, and evolution,—these operations being practically performed by addition and subtraction. For this purpose, it is necessary to have a Table of Logarithms, 80 constructed that we may readily obtain the logarithm of any number within a certain limit, or the number corresponding to any logarithm, to a certain degree of approximation. The common tables give the logarithms of numbers from 1 to 10,000, correct to 6 decimal places. With a table of this kind, we have the following obvious RULES FOR COMPUTATION. I. To multiply one number by another :-Find the logarithms of the given numbers; add these logarithms, and find the number corresponding to the sum ; this number will be the required product ; (404, 3). II. To divide one number by another :—Find the logarithms of the given numbers ; subtract the logarithm of the divisor from that of the dividend, and find the number corresponding to the difference; this number will be the required quotient; (404, 4). III. To raise a number to any power :-Find the logarithm of the given number, and multiply it by the exponent of the required power ; then find the number corresponding to this product, and it will be the required power ; (404, 5). IV. To extract any root of a number :—Find the logarithm of the given number, and divide it by the index of the root; then find the number corresponding to the quotient, and it will be the required root ; (404, 6). NOTE.–From (400), we infer that negative numbers, as such, have no logarithms. But we may always employ logarithms in calculations where negative factors are involved, by disregarding signs until the absolute value of the product or quotient is obtained. THE COMMON SYSTEM. = 406. Any positive number except unity may be made the base of a system of logarithms. But the only base used in practical calculations, is 10. The logarithms of numbers according to this base, form what is called the Common System of logarithms. NOTE.-Besides the common system, there is another, called the Nuperian System, from Baron Napier, the inventor of logarithms. This system is of great theoretical importance, and its relation to other systems will be shown in a subsequent article. 407. The peculiarities which constitute the advantage of the common system, may be shown as follows : Since 10 is the base of the system, log. 1 = log. 10° = 0, log. 10 = log. 10' 1, log. 100 log. 10' = 2, log. 1000. = log. 10° = 3, log. 10000 = log. 10* 4. Now it is obvious that if any number, integral or mixed, be greater than 1 and less than 10, its logarithm will be entirely decimal ; if the number be greater than 10 and less than 100, its logarithm will be 1 plus a decimal ; if greater than 100 and less than 1000, its logarithm will be 2 plus a decimal; and so on. Hence, 1.—The common logarithm of an integer or a mixed number will have a positive index, equal to the number of integral places minus 1. Again, since the logarithm of 10 is 1, it follows that if a number be divided by 10 continually, the logarithm will be diminished by 1 continually, the decimal part remaining unchanged. Let us take any number, as 5468, and denote the mantissa, or the decimal part of its logarithm, by m. Then we have (1.) (2.) log. 5168 log. .5468 = -1+m, log. 546.8 log. .05468 log. 54.68 log. .005468 log. 5.468 log. .0005468 3+m, -2+m, -3+m, -4+m; in which 3, 2, 1, 0, are the indices of the logarithms in column (1); and -1, -2, -3, --4, are the indices of the logarithms in columri (2); and m, the decimal part in all. Hence, 2.-1f two numbers consist of the same figures, and differ only in the position of the decimal point, their logarithms, in the common system, will have the same decimal part, and will disfer only in the values of the index. 3.— The common logarithm of a decimal fraction will have a negative index ; if the significant part of the decimal commence at the tenths' place, the index of the logarithm will be -1; but if ciphers occur between the decimal point and the first significant figure, the index of the logarithm will be numerically equal to the number of intervening ciphers, plus 1. 408. In writing the logarithm of a decimal fraction, the minus sign is placed before the index, and the decimal or positive part annexed without any intervening sign. Thus, from a table of logarithms, we have log. .0546 = -2.737193, in which the minus sign must be understood as affecting only the index 2. This logarithm is therefore equivalent to -2+.737193. COMPUTATION OF LOGARITHMS. 409. Since the rules for computing by logarithms require a logarithmic table, it becomes necessary to calculate the logarithms of an extended series of numbers. The only practical method of doing this, is by means of a converging series, expressing the valuc of any logarithm in known terms. Let us resume the fundamental equation, a = b, (1) in which w is the logarithm of b, to the base a. Assume a=1+c, b=l+p; then (1+0=1+p, (2) where x is the logarithm of 1+p, to the base a. c+ . p+.... ale+ co + .....) 3 Raise both members of equation (2) to the nth power ; then (1+c)** = (1+p)" Expanding both members by the Binomial Theorem, we have nx(nx——1) nx(nx-1)(nx-2) 1+nxc+ + 2 2 3 nx(nx—1)(nx—2)(nx— -3) n(n-1) -c+....=1+np+ - 1)p+ 2 3 4 2 n(n-1)(n —2) n(n-1)(1-2)(n-3) -p+ 2 3 2 3 4 Dropping unity from both members, and dividing by n, we obtain (n.x—1) (nx—1)(nx—2) (nx-1)(nx-2)(nx-3) ca + C + 2 , 3 , 4 -p+ p*+... 2 2 . 3 2 3 - 4 This equation is true for all values of n; it will be true, therefore, when n=0. Making this supposition, the equation reduces to ca CS p' + + 3 ...(8) -P- 3 4 5 From equation (2), we perceive that x = log (1+p). Hence, if we place 1 M= c-1c*+16—7c+{c—.... equation (3) will become pe log. (1+p) = Mp (A) 3 4 5 Thus, we have obtained an expression for the logarithm of the number 1+p, or b. This expression consists of two faotors; namely, the quantity in the parenthesis, which depends upon the number, and the quantity M, which depends upon the base of the system. 410. It is obvious that if a definite value be given to M, the base of the system will be fixed and determinate. Baron Napier arbitrarily assumed M=1. To determine the base of the system, according to this assumption, substitute 1 for M in equation (4); (409). We shall have, after reducing, + |