Elements of Algebra

66 ELEMENTS SECT. I. rithm of $c$, that is, $log. c$, and next taking the half of that logarithm, or $\frac{1}{2}log. c$, we should have the logarithm of the square root required: we have therefore only to look in the Tables for the number answering to that logarithm, in order to obtain the root required. 230. We have already seen, that the numbers, $1, 2, 3, 4, 5, 6$, &c. that is to say, all positive numbers, are logarithms of the root $a$, and of its positive powers; consequently, logarithms of numbers greater than unity: and, on the contrary, that the negative numbers, as $-1, -2$, &c. are logarithms of the fractions $\frac{1}{a}, \frac{1}{a^2}$, &c. which are less than unity, but yet greater than nothing. Hence, it follows, that, if the logarithm be positive, the number is always greater than unity: but if the logarithm be negative, the number is always less than unity, and yet greater than $0$; consequently, we cannot express the logarithms of negative numbers: we must therefore conclude, that the logarithms of negative numbers are impossible, and that they belong to the class of imaginary quantities. 231. In order to illustrate this more fully, it will be proper to fix on a determinate number for the root $a$. Let us make choice of that, on which the common Logarithmic Tables are formed, that is, the number $10$, which has been preferred, because it is the foundation of our Arithmetic. But it is evident that any other number, provided it were greater than unity, would answer the same purpose: and the reason why we cannot suppose $a=1$, is manifest; because all the powers $a^n$ would then be constantly equal to unity, and could never become equal to another given number, $c$. CHAP. XXII. Of the Logarithmic Tables now in use. 232. In those Tables, as we have already mentioned, we begin with the supposition, that the root $a=10$; so that the logarithm of any number, $c$, is the exponent to which we must raise the number $10$, in order that the power resulting from it may be equal to the number $c$; or if we denote the logarithm of $c$ by $L.c$, we shall always have $10^{L.c} = c$.