2. Divide 21.754 by 2.4678, by logarithms. Nos. Logs. 21.754 1.3375391 2.4678 0.3923100 . . 3. Divide 4.6257 by .17608, by logarithms. Nos. Logs. 4.6257 0.6651725 .17608 1.2457100 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. Nos. Logs. .27684 7.4422288 5.1576 0.7124477 . Quot. .0536761. . 2.7297811 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. Nos. Logs. 6.9875 0.8443218 .075789 2.8796062 . . Quot. 92.1967 . 1.9647156 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. Logs. .19876 1.2983290 .0012345 3.0914911 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 8. Divide 1728.95 by 1.10678, by logarithms. Ans. 1562.144 9. Divide .1023674 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, add their com plements 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. Log. of 37.125 1.56.96665 Complement 8.4303335 1.1693217 2.1312304 4ns, 53.81099 1.7308856 2. Find a fourth proportional to 05764, .7186, and .34721, by logarithms. Log. of .05764 2.7607240 3. Find a third proportional to 12 796 and 3.24718, by logarithms. Log. of 12.796 1.1070742 . 4. Find the interest of 2791. 58. for 274 days, at 45 per cent. per annum, by logarithms. Comp log of 100 8.0000000 Comp. log. of 366 7.4377071 Log. of 275.25 2.4459932 Log. of 274 . 2.4377506 Log. of 4.5. 0.6532125 . . 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 proportional to 12.796 and 3.24718 by logarithms. Ans. 17.55623 and .8240216 INVOLUTION, OR THE RAISING OR POWERS BY LOGARITHMS. Take out the logarithm of the given nu bet from the tables, and multiply it by the index of he proposed power ; the 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 aswhat is to be carried from the decimal part will be affirmative, the index of the result must be takén accord. ingly. EXAMPLES 1. Find the square of 2.7500, logarithms. Log. of 2.7568 0.4402477 2 3. Find the cube of 7.0851, by logarithms. Log, of 7.0851 0.8503399 3 |