Example 4. Required the product of 27.63, 1.859, .7258 and 0.3591. Two numbers being given, to find how many times one is contained in the other, by Logarithms. RULE. From the Logarithm of the Dividend subtract, the Logarithm of the Divisor, and the remainder will be the Logarithm, whose corresponding natural number will be the Quotient required. In this operation, the Index of the Divisor must be changed from affirmative to negative, or from negative to affirmative; and then the difference of the affirmative and negative Indices must be taken for the index to the Logarithm of the Quotient. Likewise when one has been borrowed in the left hand place of the Decimal part of the Logarithm, add it to the Index of the Divisor, if affirmative; but subtract it, if negative; and let the Index, thence arising, be changed and worked with, as before. Example 3. Divide .05951 by .007693. Example 4. Divide .6651 by 22.5. Log. of .6651-1.822887 Log. of 22.5 = Quotient 1.352183 .02956=-2.470704 PROPORTION, Or the Rule of Three in Logarithms. RULE. Having stated the three given terms according to the rule in common Arithmetic, write them orderly under one another, with the signs of proportion; then add the Logarithms of the second and third terms together, and from their sum subtract the Logarithm of the first term, and the remainder will be the Logarithm of the fourth term, or An swer. 'Or,-add together the Arithmetical Complement of the Logarithm of the first term, and the Logarithms of the second and third terms; the sum, rejecting 10 from the Index, will be the Logarithm of the fourth term, or term required. N. B. The Arithmetical Complement of a Logarithm is what it wants of 10,000000, or 20,000000, and the easiest way to find it is to begin at the left hand, and subtract every figure from 9, except the last, which should be taken from 10; but if the index exceed 9, it must be taken from 19.-It is frequently used in the rule of Proportion and Trigonometrical calculations, to change Subtrac tions into Additions. EXAMPLES. Ist. If a clock gain 14 seconds in 5 days 18 hours, how much will it gain in 17 days 15 hours? : Log.=0.759668 5.75 days Or thus; 5.75 days: Arith. Co. Log.=9,240332 17.625 :: Log.-1.246129 Answer 42". 91 -1.632589 F 2d. Find a fourth proportional to 9.485, 1.969 and 347.2. Answer 6.944 -0.841610 3d. What number will have the same proportion to .8538 as .3275 has to .0131 .0131 : Log.---2.117271 To find any proposed power of a given number by Logarithms. Rule. Multiply the Logarithm of the given number by the Index of the proposed power, and the product will be the Logarithm, whose natural number is the power required. When a negative Index is thus multiplied, its product is negative, but what was carried from the decimal part of the Logarithm must be affirmative; consequently the difference is the Index of the product, which difference must be considered of the same kind with the greater, or that which was made the minuend. EXAMPLES., 1. What is the second power of 3.874? Log. of 3.874-0.588160' Index Power required=15.01 =1.176320 2. Required the third power of the number 2.768. Log. of 2.768=0.442166 Index Answer 21.21=1.326498 3. Required the second power of theļnumber .2857. Log. of .2857=-1.455910 4. Required the third power of the number .7916. Log. of .7916=-1.898506 |