DIVISION BY LOGARITHMS. From the logarithm of the dividend, as found in the tables, subtract the logarithm of the divisor, and the natural number answering to the remainder will be the quotient required. Observing, if the subtraction cannot be made in the usual way, to add, as in the former rule, the 1 that is to be carried from the decimal part, when it occurs, to the index of the logarithm of the divisor, and then this result, with its sign changed, to the remaining index, for the index of the logarithm of the quotient. Here1, in the lower index, is changed into +1, which is then taken for the index of the result. 4. Divide .27684 by 5.1576, by logarithms. Nos. .27684 5.1576 Quot. .0536761 Here the 1 that is to be carried from the decimals, is taken as 1, and then added to - - 1, in the upper index, which gives-2 for the index of the result. Here the 1, that is to be carried from the decimals, is added to 2, which makes 1, and this put down, with its sign changed, is +1. 6. Divide .19876 by .0012345, by logarithms. Nos. .19876 .0012345 Quot. 161.0051 Here 3, in the lower index, is changed into +3, and this added to -1, the other index, gives +3 – 1 or 2. 7. Divide 125 by 1728, by logarithms. Ans. .0723379 3. Divide 1728.95 by 1.10678, by logarithms. Ans. 1562.144 9, Divide 10.23674 by 4.96523, by logarithms. Ans. 2.061685 10. Divide 19956.7 by .048235, by logarithms. Ans. 413739 11. Divide .067859 by 1234.59, by logarithms. Ans. .0000549648 THE RULE OF THREE, OR PROPORTION, BY LOGARITHMS. For any single proportion, add the logarithms of the second and third terms together, and subtract the logarithm of the first from their sum, according to the foregoing rules; then the natural number answering to the result will be the fourth term required. But if the proportion be compound, add together the logarithms of all the terms that are to be multiplied, and from the result take the sum of the logarithms of the other terms, and the remainder will be the logarithm of the term sought. Or, the same may be performed more conveniently thus, Find the complement of the logarithm of the first term of the proportion, or what it wants of 10, by beginning at the left hand, and taking each of its figures from 9, except the last significant figure on the right, which must be taken from 10; then add this result and the logarithms of the other two terms together, and the sum, abating 10 in the index, will be the logarithm of the fourth term, as before. And, if two or more logarithms are to be subtracted, as in the latter part of the above rule, and their complements and the logarithms of the terms to be multiplied together, and the result, abating as many 10's in the index as there are logarithms to be subtracted, will be the logarithm of the term required; observing when the index of the logarithm, whose complement is to be taken, is negative, to add it, as if it were affirmative, to 9; and then take the rest of the figures from 9, as before. EXAMPLES. 1. Find a fourth proportional to 37.125, 14.768, and 135.279, by logarithms. 2. Find a fourth proportional to .05764, .7186, and .34721, by logarithms. 3. Find a third proporrional to 12.796 and 3.24718, by 4. Find the interest of 2791. 53. for 274 days, at 41 per cent. per annum, by logarithms. 2 5. Find a fourth proportional to 12.678, 14.065, and 100.979, by logarithms. Ans. 112.0263 6. Find a fourth proportional to 1.9864, .4678, and 50.4567, by logarithms. Ans. 11.88262 7. Find a fourth proportional to .09658, .24958, and .008967, by logarithms. Ans. .02317234 8. Find a mean proportional between 498621 and 2.9587, and a third porportional to 12.796 and 3.24718 by logarithms. Ans. 17.55623 and .8240216. INVOLUTION, OR THE RAISING OF POWERS BY LOGARITHMS. Take out the logarithm of the given number from the tables, and multiply it by the index of the proposed power; then the natural number answering to the result, will be the power required. Observing, if the index of the logarithm be negative, that this part of the product will be negative; but as what is to be carried from the decimal part will be affirmative, the index of the result must be taken accordingly. EXAMPLES. 1. Find the square of 2.7568, by logarithms. |