Which logarithm is also correct to the nearest unit in the last figure. And in the same way we may proceed to find the logarithm of any prime number Also, because the sum of the logarithms of any two numbers gives the logarithm of their product, and the difference of the logarithms the logarithin of their quotient, &c. ; we may readily compute, from the above two logarithms, and the logarithm of 10, which is 1, a great number of other logarithms, as in the following examples : 3. Because 2 X2=4, therefore log. 2 .301029995 Mult, by 2 2 gives log. 4 .602059990 4. Because 2X3=6, therefore to .301029995 } gives log. 6 .778151250 5. Because 23=8, therefore log. 2 .301029995 mult. by 3 3 gives log. 8 .903089985 6. Because 32 =9, therefore log. 3 .477121255 mult. by 2 2 And thus, by computing, according to the general formula, the logarithms of the next succeeding prime numbers 7, 11, 13, 17, 19, 23, &c. we can find, by means of the simple rules, before laid down for multiplication, division and the raising of powers, as many other logarithms as we please, or may speedily examine any logarithm in the table. 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 logarith tive, and must, therefore, be added tegral parts, after the manner of quantities in algebra. Which method will be found mu to those who possess a slight knowl than that of using the arithmetical o 1. Multiply 37.153 by 4.086, by Nos. 37.153 1.56 4.086 0.61 . Prod. 151.8071 2. Multiply 112.246 by 13.958, I 2.04 13.958 1.14 Prod. 1563.128 3. Multiply 46.7512 by .3275, by Lo 1.669 .3275 1.5152 Prod. 15.31102 Here, the + 1, that is to be carried from cancels the – 1, and consequently there re the upper line to be set down. 4. Multiply .37816 by .04782, by logarithms. Logs. 1.5776756 ..04782 2.6796096 Here the + 1 that is to be carried from the decimals, 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. Logs. 0.576 ! 109 2.053 0.3123889 .007693 3.8860997 Here the +1, that is 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. Nos, Logs. 3.586 0.554610 2.1046 0.323170 .8372 1.922829 .0294 2.468347 Here the +2, that is to be carried, cancels the -2, and there remains the I to be set down. 7. Multiply 23.14 by 5.062 by logarithms. Ans. 117.1347 8. Multiply 4.0763 by 9.8432, by logarithms. Ans. 40.12383 9. Multiply 498.256 by 41.2467, by logarithms. Ans. 20551.41 10. Multiply 4.026747 by .012345, by logarithms. Ans. .0497102 11. Multiply 3.12567, .02868, and .12379, together, by logarithms. Ans. 09109705 12. Multiply 2876.9, 10674, .098762, apd .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. EXAMPLES. 1. Divide 4768.2 by 36.954, by logarithms. Logs. 3.6783545 36.954 1.5676615 . |