Elements of Algebra

CHAP. XXIII. OF ALGEBRA. 75 together the logarithms of those numbers, the calculation will be as follows: $log. 343 = 2.5352941$ } added $log. 2401 = 3.3803922$ } —————— $5.9156863$ their sum $log. 823540 = 5.9156847$ nearest tabular log. —————— $16$ difference, which in the Table of Differences answers to $3$; this therefore being used instead of the cipher, gives $823543$ for the product sought: for the sum is the logarithm of the product required; and its characteristic $5$ shews that the product is composed of $6$ figures; which are found as above. 255. But it is in the extraction of roots that logarithms are of the greatest service; we shall therefore give an example of the manner in which they are used in calculations of this kind. Suppose, for example, it were required to extract the square root of $10$. Here we have only to divide the logarithm of $10$, which is $1.0000000$ by $2$; and the quotient $0.5000000$ is the logarithm of the root required. Now, the number in the Tables which answers to that logarithm is $3.16228$, the square of which is very nearly equal to $10$, being only one hundred thousandth part too great*. * In the same manner, we may extract any other root, by dividing the log. of the number by the denominator of the index of the root to be extracted; that is, to extract the cube root, divide the log. by $3$, the fourth root by $4$, and so on for any other extraction. For example, if the 5th root of $2$ were required, the log. of $2$ is $0.3010300$: therefore $5)0.3010300$ —————— $0.0602060$ is the log. of the root, which by the Tables is found to correspond to $1.1497$; and hence we have $\sqrt[5]{2} = 1.1497$. When the index, or characteristic of the log. is negative, and not divisible by the denominator of the index of the root to be extracted; then as many units must be borrowed as will make it exactly divisible, carrying those units to the next figure, as in common division.