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. 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. 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. 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; log. x 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. |