arithm, must not be confounded, with the sign which denotes that the natural number is negative. That which the index of the logarithm is intended to show, is not whether the natural number is positive or negative, but whether it is greater or less than a unit. (Art. 16.) In these examples, the logarithms are taken from the tables, and added, in the same manner, as if both factors were positive. But after the product is found, the negative sign is prefixed to it, because + is multiplied into (Alg. 105.) Here, the indices of the logarithms are negative, but the product is positive, because the factors are both positive. DIVISION BY LOGARITHMS. 41. FROM THE LOGARITHM OF THE DIVIDEND, SUBTRACT THE LOGARITHM OF THE DIVISOR; THE DIFFERENCE will be the LOGARITHM OF THE QUOTIENT. (Art. 36.) 42. The decimal part of the logarithm may be subtracted as in common arithmetic. But for the indices, when either of them is negative, or the lower one is greater than the upper one, it will be necessary to make use of the general rule for subtraction in algebra; that is, to change the signs of the subtrahend, and then proceed as in addition. (Alg. 82.) When 1 is carried from the decimal part, this is to be considered affirmative, and applied to the index, before the sign is changed. In this example, the upper logarithm being less than the lower one, it is necessary to borrow 10, as in other cases of subtraction; and therefore to carry 1 to the lower index, which then becomes +2. This changed to -2, and added to - 1 above it, makes the index of the difference of the logarithms - 3. Here, 1 carried to the lower index, makes it +4. This changed to 4, and added to 1 above it, gives - 3 for the index of the difference of the logarithms. The quotient of 0.0985 divided by 0.007241, is 13.6. 43. To divide negative quantities, proceed in the same manner as if they were positive, (Art. 40.) and prefix to the quotient, the sign which is required by the rules for division in algebra. In these examples, the sign of the divisor being different from that of the dividend, the sign of the quotient must be negative. (Alg. 123.) 44. Involving a quantity is multiplying it into itself. By means of logarithms, multiplication is performed by addition. If, then, the logarithm of any quantity be added to itself, the logarithm of a power of that quantity will be obtained. But adding a logarithm, or any other quantity, to itself, is multiplication. The involution of quantities, by means of logarithms, is therefore performed, by multiplying the logarithms. 4 =2×2. =2X3. of 100 x 100 x 100 x 100, 100* is 2+2+2+2 =2×4. On the same principle, the logarithm of 100" is 2 xn. 45. To involve a quantity by logarithms. MULTIPLY THE LOGARITHM OF THE QUANTITY, BY THE INDEX OF THE POWER REQUIRED. The reason of the rule is also evident, from the consideration, that logarithms are the exponents of powers and roots, and a power or root is involved, by multiplying its index into the index of the power required. (Alg. 220, 288.) 46. It must be observed, as in the case of multiplication, (Art. 38.) that what is carried from the decimal part of the logarithm is positive, whether the index itself is positive or negative. Or, if 10 be added to a negative index, to render it positive, (Art. 12.) this will be multiplied, as well as the other figures, so that the logarithm of the square, will be 20 too great; of the cube, 30 too great, &c. |