Negative numbers can be divided in the same manner as positive, taking care to prefix to the quotient the proper sign, according to Art. 53. 251. Instead of subtracting one logarithm from another, it is sometimes more convenient to add what it lacks of 10, and from the sum reject 10. The result is evidently the The remainder found by subtracting a logarithm from 10 is called the arithmetical complement of the logarithm, or the cologarithm. The cologarithm is easiest found by beginning at the left of the logarithm, and subtracting each figure from 9, except the last significant figure, which must be subtracted from 10. By this method, Ex. 1 will appear as follows: 252. In working examples combining multiplication and division, the use of cologarithms is of great advantage. RULE. Find the sum of the logarithms of the multipliers and the cologarithms of the divisors; reject as many tens as there are cologarithms (divisors); the result will be the logarithm of the number sought. An even number of negative quantities gives a positive result, an odd number a negative (Art. 48). 253. Add the cologarithm of the first term to the logarithms of the second and third terms, and from the sum reject 10. 1. Given 14:175=7486 : x, to find x. Ans. 359.3+ 2. Given 416 584256: x, to find x. 3. Given x 17949.68: 489, to find x. Ans. 18.18+ INVOLUTION BY LOGARITHMS. RULE. 254. Multiply the logarithm of the number by the exponent of the power required (Art. 126). In involution, as the error in the logarithm is multiplied by the index of the power, the results with logarithms of only four decimal places cannot be relied on for more than two or three significant figures. 1. Find the 15th power of 1.17. 3. Find the 4th power of 0.0176. Ans. 0.00000009595. 4. Find the 9th power of 1.179. Ans. 4.49. Negative numbers are involved in the same manner, taking care to prefix to the power the proper sign, according to Art. 125. 5. Find the 3d power of -0.017. Ans. -0.000004913. 6. Find the 6th power of -14. Ans. 7529536. In the last two examples the exact answers are given, though from the table only answers approximating to these can be obtained. ፡፡ EVOLUTION BY LOGARITHMS. RULE. 255. Divide the logarithm of the number by the exponent of the root required (Art. 143). When the characteristic is negative, and not divisible by the index of the root, we increase the negative characteristic so as to make it divisible, and to the mantissa prefix an equal positive number. 1. Find the 5th root of 0.0173. Negative numbers are evolved in the same manner, taking care to prefix to the root the proper sign, according to Art. 136. 4. Find the 7th root of -17. 5. Find the 5th root of -0.00496. Ans. 1.499. Ans. 0.346. SYSTEMS OF LOGARITHMS. 257. The system of logarithms which has 10 for its base is the one in common use. As in this system the mantissa of the logarithm of any set of figures is the same, wherever the decimal point may be (Art. 243), which (in the Arabic notation of numbers) would not be the case with any other base, it is far the most convenient system. The number of possible systems, however, is infinite. In general, if a*=n, then x is the logarithm of n to the base a; and n is the number (sometimes called the antilogarithm) corresponding to the logarithm x, in a system whose base is a. 258. The logarithm of 1 is 0, whatever the base may be. For the 0 power of every quantity is 1, or ao1 (Art. 120). 259. The logarithm of the base itself is 1. For the first power of any quantity is that quantity itself, or a1(Art. 119). 260. Neither 0 nor 1 can be the base of a system of logarithms. For all the powers and roots of 0 are 0, and of 1 are 1. 261. The logarithm of the reciprocal of any quantity is the negative of the logarithm of the quantity itself. For the reciprocal of any quantity is 1 divided by that quantity (Art. 27); that is, is the logarithm of 1 minus the logarithm of the quantity; or 0 minus the logarithm of the quantity (Art. 250). 262. In a system whose base is greater than 1, the logarithm of infinity (∞) is infinity; and the logarithm of 0 is minus infinity (—∞). Hence, negative quantities cannot have logarithms. 263. In a system whose base is between 1 and 0, the less the number the greater its logarithm. For the greater the power of a proper fraction the less its |