100 is 2+2+2 4 =2×2. =2×3. of 100 x 100 x 100 × 100, 100* is 2+2+2+2=2×4. On the same principle, the logarithm of 100" is 2×n. 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.) Ex. 1. What is the cube of 6.29? Root 6.296, its log. 0.79906 Index of the power Power 249.6 3 2.39718 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. Ex. 1. Required the cube of 0.06-49 Root 0.0649 log. 2.81224 Index Power 0.0002733 or 8.81224 3 3 The index -2, multiplied by 3, becomes-6. But +2 carried from the decimal part, reduces it to -4. Or, if 10 be added to the index, so as to make it +8; when this is multiplied by 3, and the product increased by 2 carried from the decimal part, it becomes 26, which is 30 too great. Leaving off the first figure, it still remains 10 too great, +6 instead of -4. 2. Required the 4th power of 0.1234 47. Evolution is the opposite of involution. Therefore, as quantities are involved, by the multiplication of logarithms, roots are extracted by the division of logarithms; that is, To extract the root of a quantity by logarithms, Divide the logarithm of the quantity, by the number expressing the root required. The reason of the rule is evident also, from the fact, that logarithms are the exponents of powers and roots, and evolution is performed, by dividing the exponent, by the number expressing the root required. (Alg. 257.) 1. Required the square root of 648.3. In the first of these examples, the logarithm of the given number is divided by 2; in the other, by 3. 3. Required the 10th root of 6948. Power 6948 10)3.84186 Root 2.422 0.38418 The division is performed here, as in other cases of decim als, by removing the decimal point to the left. 5. What is the ten thousandth root of 49680000? We have, here, an example of the great rapidity with which arithmetical operations are performed by logarithms. E 48. If the index of the logarithm is negative, and is not divisible by the given divisor, without a remainder, a difficulty will occur, unless the index be altered. Suppose the cube root of 0.0000892 is required. The logarithm of this is 5.95036. If we divide the index by 3, the quotient will be 1, with -2 remainder. This remainder, if it were positive, might, as in other cases of division, be prefixed to the next figure. But the remainder is negative, while the decimal part of the logarithm is positive; so that, when the former is prefixed to the latter, it will make neither +2.9 nor -2.9, but -2+.9. This embarrassing intermixture of positives and negatives may be avoided, by adding to the index another negative number, to make it exactly divisible by the divisor. Thus, if to the index -5: there be added-1, the sum -6 will be divisible by 3. But this addition of a negative number must be compensated, by the addition of an equal positive number, which may be fixed to the decimal part of the logarithm. The division may then be continued, without difficulty, through the whole. pre Thus, if the logarithm 5.95036 be altered to 6+1.95036 it may be divided by 3, and the quotient will be 2.65012. We have then this rule, 49. Add to the index, if necessary, such a negative number as will make it exactly divisible by the divisor, and prefix an equal positive number to the decimal part of the logarithm.. 50. If, for the sake of performing the division conveniently, the negative index be rendered positive, it will be expedient to borrow as many tens, as there are units in the number denoting the root. Here the index, by borrowing, is made 40 too great, that is, +38 instead of -2. When, therefore, it is divided by 4,. it is still 10 too great, +9 instead of -1. What is the 5th root of 0.008926? Power 0.008926 Root 0.38916 5)3.95066 or '5)47.95066 9.59013 51. A power of a root may be found by first multiplying the logarithm of the given quantity into the index of the power, (Art. 45.) and then dividing the product by the number expressing the root. (Art. 47.) 1. What is the value of (53), that is, the 6th power of the 7th root of 53? 2. What is the 8th power of the 9th root of 654 ? PROPORTION BY LOGARITHMS. 52. In a proportion, when three terms are given, the fourth is found, in common arithmetic, by multiplying together the second and third, and dividing by the first. But, when logarithms are used, addition takes the place of multiplication, and subtraction, of division. To find then, by logarithms, the fourth term in a proportion, ADD the logarithms of the SECOND and THIRD terms, and from the sum SUBTRACT the logarithm of the FIRST term. The remainder will be the logarithm of the term required. Ex. 1. Find a fourth proportional to 7964, 378, and 27960. |