Then, as 127: 100 :: 81: 64 nearly, the numeral difference. Therefore 34'09264 is the number sought, two integers being marked off, because the index of the given logarithm is 1. Had the index been negative, thus, r5326606, its corresponding number would have been 3409264, wholly decimal. Or, the proportional numeral difference may be found, in the best tables, by inspection of the small tables of proportional parts, placed in the margin. MULTIPLICATION BY LOGARITHMS. RULE. Take out the logarithms of the factors from the table, then add them together, and their sum will be the logarithm of the product required. Then, by means of the ta ble, take out the natural number answering to the sum, for the product sought. NOTE I. In every operation, what is carried from the decimal part of a logarithm to its index is affirmative; and is therefore to be added to the index, when it is affirmative; but subtracted, when it is negative. NOTE 2. When the indices have like signs, that is, both or both, they are to be added, and the sum has the common sign; but when they have unlike signs, that is, one and the other, their difference, with the sign of the greater, is to be taken for the index of the sum. EXAMPLES. 3. To multiply 3'902, 59716 and 0314728 all to Here the 2 cancels the 2, and the 1, to be carried from the decimals, is set down. 4. To multiply 3'586, 21046, 08372 and o'0294 all together. - Here the 2, to be carried, cancels the 2, and there remains the I to be set down. DIVISION BY LOGARITHMS. RULE. From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required. NOTE. If I be to be carried to the index of the subtrahend, apply it according to the sign of the index; then change the sign of the index to -, if it be +, or to +, if it be ; and proceed according to the second note un-. der the last rule. Here 1, carried from the decimals to the -3, makes it become-2, which, taken from the other --2, leaves remaining. 4. To divide 7438 by 12'9476. Here the 1, taken from the 1, makes it become to be set down. INVOLUTION BY LOGARITHMS. RULE. Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required. NOTE. A negative index, multiplied by an affirmative number, gives a negative product; and as the number carried from the decimal part is affirmative, their difference with the sign of the greater is, in that case, the index of the product. EXAMPLES |