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. Here — 1, 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. 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. 5. Divide 6.9875 by .075789, by logarithms. to Here the 1, that is to be carried from the decimals, is added 2, which makes changed, is +1. 1, and this put down, with its sign 6. Divide .19876 by .0012345, by logarithms. Nos. .19786 . .001.2345 Logs. 1.2983290 3.0914911 Here Quot. 161.0051 . 2.2068379 3, in the lower index, is changed into +3, and 1, the other index, gives + 3 — 1 or 2. this added to Ans. 0723379. 7. Divide 125 by 1728, 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, 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. Bu 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 most 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 lefthand, 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, add their complements and the logarithms of the terms to be multiplied together, and the result, abating as many 10s 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 proportion to 37.125, 14.768, and 135,279, by logarithms. Log. of 37.125 Ans. 53.81099 1.5696665 8.4303335 1.1693217 2.1312304 1.7308856 2. Find a fourth proportional to .05764, .7186, and .34721, 3. Find a third proportional to 12.796, and 3.24718, by Log. of 3.24718 0.5115064 1.9159386 Ans. .8240216 4. Find the interest of 2797. 5s. for 274 days, at 41 per cent. per annum, by logarithms. Comp. log. of 100 Comp. log. of 365 Log. of 279.25 Log. of 4.5 Ans. 9.433296 '. 8.0000000 7.4377071 2.4459932 2.4377506 0.6532125 0.9746634 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 .005967, by logarithms. Ans. .02317234. 8. Find a third proportional to .498621, and 2.9587, and a third proportional 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. Log. of 2.7568 Square 7.599946 0.4402477 0.8804954 2. Find the cube of 7.0851, by logarithms. 3 Find the fifth power of .87451, by logarithms. Log. of .87451 1.9417648 5 Fifth power .5114695. 1.7088240 Where 5 times the negative index 1, being +4 to carry, the index of the power is 1. 4. Find the 365th power of 1.0045, by logarithms. Log. 1.0045* 0.0019499 5, and 5. Required the square of 6.05987, by logarithms. Ans. 36.72603. 6. Required the cube of .176546, by logarithms. Ans. 005502674. 7. Required the 4th power of .076543, by logarithms. 8. Required the 5th power of 2.97643, 9. Required the 6th power of 21.0576, Ans. 0000343259. by logarithms. Ans. 233.6031. by logarithms. Ans. 87187340. 10. Required the 7th power of 1.09684, by logarithms. Ans. 1.909864. * The answer, 5.148888, though found strictly according to the general rule, is not correct in the last four figures, 8888; nor can the answers to such questions, relating to very high powers, be generally found true to 6 places of figures by the tables of Log. commonly used; if any power above the hundred thousandth were required, not one figure of the answer here given could be depended on. The Log. of 1.0045 is 00194994108 true to eleven places, which, multiplied by 365, gives .7117285 true to 7 places, and the corresponding number true to 7 places is 5.149067. See Doctor Adrain's edition of Hut. Math. Vol. 1. p. 169. |