Elements of Algebra

68 ELEMENTS SECT. I. Let us try another fraction, which, in consequence of what we have already found, must be contained between $\frac{2}{7}$ and $\frac{1}{3}$. Such a fraction between these limits is $\frac{3}{10}$; and it is therefore required to find whether $10^{\frac{3}{10}} = 2$; if this be the case, the tenth powers of those numbers are also equal: but the tenth power of $10^{\frac{3}{10}}$ is $10^3 = 1000$, and the tenth power of 2 is 1024; we conclude therefore, that $10^{\frac{3}{10}}$ is less than 2, and, consequently, that $\frac{3}{10}$ is too small a fraction; and therefore the log. 2, though less than $\frac{1}{3}$, is yet greater than $\frac{3}{10}$. 236. This discussion serves to prove, that log. 2 has a determinate value, since we know that it is certainly greater than $\frac{3}{10}$, but less than $\frac{1}{3}$; we shall not however proceed any farther in this investigation at present. Being therefore still ignorant of its true value, we shall represent it by $x$, so that $log. 2 = x$; and endeavour to shew how, if it were known, we could deduce from it the logarithms of an infinity of other numbers. For this purpose, we shall make use of the equation already mentioned, namely, $log. cd = log. c + log. d$, which comprehends the property, that the logarithm of a product is found by adding together the logarithms of the factors. 237. First, as $log. 2 = x$, and $log. 10 = 1$, we shall have $log. 20 = x + 1$, $log. 200 = x + 2$ $log. 2000 = x + 3$, $log. 20000 = x + 4$ $log. 200000 = x + 5$, $log. 2000000 = x + 6$, &c. 238. Farther, as $log. c^2 = 2 log. c$, and $log. c^3 = 3 log. c$, and $log. c^4 = 4 log. c$, &c. we have $log. 4 = 2x$; $log. 8 = 3x$; $log. 16 = 4x$; $log. 32 = 5x$; $log. 64 = 6x$, &c. Hence we find also, that $log. 40 = 2x + 1$, $log. 400 = 2x + 2$ $log. 4000 = 2x + 3$, $log. 40000 = 2x + 4$, &c. $log. 80 = 3x + 1$, $log. 800 = 3x + 2$ $log. 8000 = 3x + 3$, $log. 80000 = 3x + 4$, &c. $log. 160 = 4x + 1$, $log. 1600 = 4x + 2$ $log. 16000 = 4x + 3$, $log. 160000 = 4x + 4$, &c. 239. Let us resume also the other fundamental equation, $log. \frac{c}{d} = log. c - log. d$, and let us suppose $c = 10$, and $d = 2$; since $log. 10 = 1$, and $log. 2 = x$, we shall have $log. \frac{10}{2}$, or $log. 5 = 1 - x$, and shall deduce from hence the following equations: