Example 2. Required the product of 46.75 and .3275. Log. of 46.75= 1.669782 Product 15,31 = 1.184993 + Here, the +1, that is to be carried from the decimals, cancels the 1, and consequently there remains 1 in the upper line to be set down. Example 3. Required the product of 3.768, 2.053 and .007693. 3.768 Log. of 0.576111 0.312389 Log. of .007693=-3.886096 Product .05951+-2.774596 In this example there is 1 to carry from the decimal part of the Logarithms, which subtracted from 3, the negative index, which leaves 2, the index of the sum of the Logarithms, and is negative. Example 4. Required the product of 27.63, 1.859, .7258 and 0.3591. Log. of 27.63 = 1.441381 Log. of 1.859 = 0.269279 Log. of .7258=-1.860817 Log. of .03591=-2.555215 Product nearly=1.339 = 0.126692 In this example there is 2 to carry from the decimal part of the Logarithms, which added to 1, the affirmative index, makes 3, from this take 3, the sum of the negative indices, the remainder is 0, which is the index of the sum of the Logarithms. 5. Required the product of 23.14 and 50.62, by Logarithms. Ans. 117.1347 6. Required the product of 3.12567, .02868 and .12379, by Logarithms. Ans. .09109705 7. Required the product of .1508, .0139 and 756.9, by Logarithms. Ans. 1.586553 8. Required the product of 637.8 and 89.27, by Logarithms. Ans. 56936.406 9. Required the product of 14 and 8.45, by Logarithms. Ans. 118.30 DIVISION. 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 Quo. tient. 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 1. Divide 558.9 by 6.48. Log. of 558.9 =2.747334 Log. of 6.48 =0.811575 Here, the 1 to be taken from the decimals is taken as 1, which when added to 2, the index of the dividend, leaves 1 for the index of the quotient; that is, 2-1 = 1. Example 2. Divide 15.31 by 46.75. 1.184975 1.669782 Quotient =.3275=-1.515193 Here, the 1 to be taken from the decimals, is added to 1, the index of the divisor makes 2; this with its sign changed is 2, from which subtracting 1, the index of the dividend, the remainder is-1, which is negative, because the negative index is greater. Example 3. Divide .05951 by .007693. Log. of .007693=-3.886096 Here, the 1 to be taken from the decimals, is subtracted from - 3, which leave - 2, this changed is + 2; and this added to-2, the other index, gives 2-2=0. Example 4. Divide .6651 by 22.5. 1.352183 Quotient-.02956=-2.470704 Here,+ 1 in the lower index, is changed into 1, and this added to 1, the other index, gives -1-1, or 2, the index of the result. 5. Required the quotient of 125 divided by 1728, by Logarithms. Ans. .0723379 6. Divide 1728.95 by 1.10678, by Logarithms. Ans. 1562.144 7. Divide .067859 by 1234.59, by Logarithms. Ans. .0000549648 8. Divide .7438 by 12.9476, by Logarithms. 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 fequently used in the rule of Proportion and Trigonometrical calculations, to change Subtractions into Additions.* EXAMPLES. 1st. If a clock gain 14 seconds in 5 days 18 hours, how much will it gain in 17 days 15 hours? 5.75 days : Log.=0.759668 Or thus; 5.75 days: Arith. Co. Log.=9.240332 17.625 :: 14 Seconds: Log.=1.246129 Log. 1.146128 =1.632589 Answer=42". 91 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? * When the index is negative add it to 9, and subtract as before. And for every arithmetical complement that is added, subtract 10 from the last sum of the indices. |