Elements of Algebra

CHAP. XXI. OF ALGEBRA. 65 we shall obtain $a^{\log. c - \log. d} = \frac{c}{d}$; and, consequently, $\log. c - \log. d = \log. \frac{c}{d}$. 226. This leads us to the two principal properties of logarithms, which are contained in the equations $\log. c + \log. d = \log. cd$, and $\log. c - \log. d = \log. \frac{c}{d}$. The former of these equations teaches us, that the logarithm of a product, as $cd$, is found by adding together the logarithms of the factors; and the latter shews us this property, namely, that the logarithm of a fraction may be determined by subtracting the logarithm of the denominator from that of the numerator. 227. It also follows from this, that when it is required to multiply, or divide, two numbers by one another, we have only to add, or subtract, their logarithms; and this is what constitutes the singular utility of logarithms in calculation: for it is evidently much easier to add, or subtract, than to multiply, or divide, particularly when the question involves large numbers. 228. Logarithms are attended with still greater advantages, in the involution of powers, and in the extraction of roots; for if $d = c$, we have, by the first property, $\log. c + \log. c = \log. cc$, or $c^2$; consequently, $\log. cc = 2 \log. c$; and, in the same manner, we obtain $\log. c^3 = 3 \log. c$; $\log. c^4 = 4 \log. c$; and, generally, $\log. c^n = n \log. c$. If we now substitute fractional numbers for $n$, we shall have, for example, $\log. c^{\frac{1}{2}}$, that is to say, $\log. \sqrt{c}, = \frac{1}{2} \log. c$; and lastly, if we suppose $n$ to represent negative numbers, we shall have $\log. c^{-1}$, or $\log. \frac{1}{c}, = - \log. c$; $\log. c^{-2}$, or $\log. \frac{1}{c^2}, = -2 \log. c$, and so on; which follows not only from the equation $\log. c^n = n \log. c$, but also from $\log. 1 = 0$, as we have already seen. 229. If therefore we had Tables, in which logarithms were calculated for all numbers, we might certainly derive from them very great assistance in performing the most prolix calculations; such, for instance, as require frequent multiplications, divisions, involutions, and extractions of roots: for, in such Tables, we should have not only the logarithms of all numbers, but also the numbers answering to all logarithms. If it were required, for example, to find the square root of the number $c$, we must first find the loga- F