MULTIPLICATION BY LOGARITHMS. Take out the logarithms of the factors from the table, and add them together; then the natural number answering to the sum will be the product required. Observing, in the addition, that what is to be carried from the decimal part of the logarithms is always affirmative, and must, therefore, be added to the indices, or integral parts, after the manner of positive and negative quantities in algebra. 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. - Here the +1 to be carried destroys the 1, in the upper line, as before, and there remains the - 2 to be set down. 5. Multiply 3.768, 2.053, and .007693, together, Here the +1 to be carried from the decimals, when added to -3, makes 2, to be set down. - 6. Multiply 3.586, 2.1046, .8372, and .0294, together. Here the +2 to be carried, cancels the there remains the to be set down. - 1 2, and 7. Multiply 4.0763 by 9.8432, by logarithms. Ans. 40.12383 8. Multiply 498.256 by 41.2467, by logarithms. Ans. 20551.41 9. Multiply 4.026747 by .012345, by logarithms. Ans. .0497102 10. Multiply 3.12567, .02868, and .12379, together, by logarithms. Ans. .01109705 11. Multiply 2876.9, .10674, .098762, and .0031598, by logarithms. Ans. .0958299 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, whch 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 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. Here the 1, that is to be carried, is added to -2, which makes 1, and this put down with its sign changed is +1. 6. Divide .19876 by .0012345, by logarithms. 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, 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 |