and if a greater degree of exactness is required, we must take a greater number of integral fractions. 227. If we suppose a to preserve a constant value in the equation whilst N is made, in a* = N, succession, equal to every possible number, it is plain that will undergo changes corresponding to those made in N. By the method explained in the last arti cle, we can determine, for each value of N, the corresponding value of x, either exactly or approximatively. The value of x, corresponding to any assumed value of the number N, is called the, logarithm of that number; and a is called the base of the system in which the logarithm is taken. Hence, The logarithm of a number is the exponent of the power to which it is necessary to raise the base, in order to produce the given number. The logarithms of all numbers corresponding to a given base constitute a system of logarithms. Any positive number except 1 may be taken as the base of a system of logarithms, and if for that particular base, we suppose the logarithms of all numbers to be computed, they will constitute what is called a system of logarithms. Hence, we see that there is an infinite number of systems of loga rithms. 228. The base of the common system of logarithms is 10, and if we designate the logarithm of any number taken in that system by log, we shall have, We see, that in the common system, the logarithm of any number between 1 and 10, is found between 0 and 1. The logarithm of any number between 10 and 100, is between 1 and 2; the logarithm of any number between 100 and 1000, is between 2 and 3; and so on. The logarithm of any number, which is not a perfect power of the base, will be equal to a whole number, plus a fraction, the value of which is generally expressed decimally. The entire part is called the characteristic, and sometimes the index. By examining the several powers of 10, we see, that if a number is expressed by a single figure, the characteristic of its logarithm will be 0; if it is expressed by two figures, the characteristic of its logarithm will be 1; if it is expressed by three figures, the characteristic will be 2; and if it is expressed by n places of figures, the characteristic will be n-1. If the number is less than 1, its logarithm will be negative, and by considering the powers of 10, which are denoted by negative exponents, we shall have, Here we see that the logarithm of every number between 1 and .1 will be found between 0 and 1; that is, it will be equal to The logarithm of any number - 3, plus a fraction, In the first case, the characteristic is between 2 and 3, and so on. 1, in the second - 2, in the third 3, and in general, the characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of O's which immediately follow the decimal point. The decimal part is always positive, and to indicate that the negative sign extends only to the characteristic, it is generally written over it; thus, log 0.0122.079181, which is equivalent to 2 + .079181. 228*. A table of logarithms, is a table containing a set of numbers, and their logarithms so arranged that we may, by its aid, find the logarithm of any number from 1 to a given number, generally 10,000. The following table shows the logarithms of the numbers, from 1 to 100. When the number exceeds 100, the characteristic of its loga rithm is not written in the table, but is always known, since it is 1 less than the number of places of figures of the given number. Thus, in searching for the logarithm of 2970, in a table of logarithms, we should find opposite 2970, the decimal part .472756. But since the number is expressed by four figures, the characteristic of the logarithm is 3. Hence, log 2970 = 3.472756, and by the definition of a logarithm, the equation . a2 = N, gives 103-472756 = 2970. General Properties of Logarithms. 229. The general properties of logarithms are entirely independent of the value of the base of the system in which they are taken. In order to deduce these properties, let us resume the equation, a* = N, in which we may suppose a to have any positive value except 1. 230. If, now, we denote any two numbers by N' and N", and their logarithms, taken in the system whose base is a, by x' and ', we shall have, from the definition of a logarithm, If we multiply equations (1) and (2) together, member by member, we get, But since a is the base of the system, we have from the definition, x+x= log(N' x N"); that is, The logarithm of the product of two numbers is equal to the sum of their logarithms. 231. If we divide equation (1) by equation (2), member by The logarithm of the quotient which arises from dividing one number by another is equal to the logarithm of the dividend minus the logarithm of the divisor. 232. If we raise both members of equation (1) to the n power, we have, anx' =N'n But from the definition, we have, nx' log (N'"); that is, = (5). The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. 233. If we extract the nth root of both members of equation (1), we shall have, The logarithm of any root of a number is equal to the loga rithm of the number divided by the index of the root. 234. From the principles demonstrated ir. the four preceding articles, we deduce the following practical rules: First, To multiply quantities by means of their logarithms. Find from a table, the logarithms of the given factors, take the sum of these logarithms, and look in the table for the cor. responding number; this will be the product required. |