Evolution by Logarithms. When the logarithm of the given number has a negative characteristic, instead of being increased by 10, it should be increased by as many times 10 as there are units in the exponent of the root, and the quotient will in this case exceed its true value by 10. 32. EXAMPLES. 1. Find the fifth root of 0.028145. Solution. We have, by the tables, 50 + log. 0·028145 = 48.44940, which, divided by 5, gives 10 + log. 0-48964 = 9.68988, and the required root is 0.48964. 2. Find the cube root of 0.002197. Ans. 0.13. 3. Find the 10th root of 0.000000001. Ans. 0.12589. 4. Find the square root of 238.149. Ans. 15.4317. 33. The arithmetical complement of a logarithm is the remainder after subtracting it from 10. 34. Corollary. The arithmetical complement of the logarithm of a number is, by art. 14, and the preceding article, the logarithm of its reciprocal increased by 10. 35. Corollary. The most convenient method of finding the arithmetical complement of a logarithm is to subtract the first significant figure on the right from 10, and each figure to the left of this figure from 9. Arithmetical Complement. 36. EXAMPLES. 1. Find the arithmetical complement of 9.62595. Ans. 0.37405. 2. Find the arithmetical complement of the logarithm of 6. Ans. 9.22185. 3. Find the arithmetical complement of the logarithm of 0.07. Ans. 11.15490. 4. Find the reciprocal of 0-01115. Solution. We have, by the tables, log. 0·01115 (ar. co.) 11.95273 10- 1.95273 and the required reciprocal is 89-686. 5. Find the reciprocal of 2330. Ans. 0.00042918. 6. Find the reciprocal of 68.99. Ans. 0.014494. 37. Problem. To find the quotient of one number divided by another by means of logarithms. Solution. Subtract the logarithm of the divisor from that of the dividend, and the number, of which the remainder is the logarithm, is, by art. 13, the required quotient. Or, since, by art. 81, multiplying by the reciprocal of a number is the same as dividing by it, add the logarithm of the dividend to the arithmetical complement of the logarithm of this divisor, and the sum diminished by 10 is the logarithm of the quotient. Division by Logarithms. When the logarithm of the dividend is written 10 more than its true value, 20 must be subtracted from the sum, instead of 10. 38. EXAMPLES. 1. Divide 0.01478 by 0.9243. Solution. We have, by the tables, 10 + log. 0·01478 8.16967 log. 0.9243 (ar. co.) 10.03419 10 + log. 0·01599 8.20386 and the required quotient is 0.01599. 39. Corollary. The value of any fraction may be found by adding together the logarithms of all the factors of the numerator and the arithmetical complements of the logarithms of all the factors of the denominator, and subtracting from the sum as many times 10 as there are arithmetical complements plus as many times 10 as there are logarithms of the factors of the numerator, which are written greater than their true value by 10; the remainder is the logarithm of the fraction. Various Examples of the use of Logarithms. 40. EXAMPLES. 1. Find the value of the fraction (0-327) X v 19.81 (1-23) x (0-005) Solution. We have, from the tables, 10+ log. (0-327)? 6.60185 log. ✓ 19.81 0-64844 log: (1-23)(ar. co.) 9-97003 log. (0·005)2 (ar. co.) 14.60206 log. 66-433 1.82238 and the required value is 66.433. 41. Corollary. The logarithm of the fourth term of a proportion is found by adding together the arithmetical complement of the logarithms of the first term and the logarithms of the second and third terms. Various Examples of the use of Logarithms. 42. EXAMPLES. 1. Find the fourth term of the proportion 963 : 1279 = 8.7: %. Solution. log. 963 (ar. co.) 7.01637 3.10687 0.93952 log. 11-555 1.06276 log. 87 and we have a = 11.555. 2. Find the fourth term of the proportion 0.0138:0•319 = 76.5 : x. Ans. x = 1768.3. 43. Problem. To solve the exponential equation a = m, by means of logarithms. Solution. The logarithms of the two members of this equation give x log. a = log. m; hence log. m log. a' or log. a = log. log. m log. log. a; that is, the root of this equation is equal to the logarithm of m divided by the logarithm of a, and this quotient may be obtained by the aid of logarithms. |