And a decimal fraction having its first figure in the 1st, 2d, 3d, 4th, &c, place of the decimals, has always -1, -2, -3, -4, &c, for the index of its logarithm. It may also be observed, that though the indices of fractional quantities are negative, yet the decimal parts of their logarithms are always affirmative. And the negative mark ( – ) may be set either before the index or over it. 1. TO FIND, IN THE TABLE, THE LOGARITHM TO ANY NUMBER*. 1. If the given Number be less than 100, or consist of only two figures; its log. is immediately found by inspection in the first page of the table, which contains all numbers from 1 to 100, with their logs. and the index immediately annexed in the next column. So the log. of 5 is 0.698970. The log. of 23 is 1.361728. The log. of 50 is 1.698970. And so on. 2. If the Number be more than 100 but less than 10000; that is, consisting of either three or four figures; the decimal part of the logarithm is found by inspection in the other pages of the table, standing against the given number, in this manner; viz. the first three figures of the given number in the first column of the page, and the fourth figure one of those along the top line of it; then in the angle of meeting are the last four figures of the logarithm, and the first two figures of the same at the beginning of the same line in the second column of the page: to which is to be prefixed the proper index, which is always 1 less than the number of integer figures. So the logarithm of 251 is 2·399674, that is, the decimal 399674 found in the table, with the index 2 prefixed, because the given number contains three integers. And the log. of 34 09 is 1·532627, that is, the decimal ⚫532627 found in the table, with the index 1 prefixed, because the given number contains two integers. 3. But if the given Number contain more than four figures; take out the logarithm of the first four figures by inspection in the table, as before, as also the next greater logarithm, subtracting the one logarithm from the other, as also their corresponding numbers the one from the other. Then say, As the difference between the two numbers, Is to the difference of their logarithms, * See the table of Logarithms, after the Geometry, at the end of this volume. Which part being added to the less logarithm, before taken out, gives the whole logarithm sought very nearly. EXAMPLE. To find the logarithm of the number 34.0926. The log. of 340900, as before, is 532627. is 532754. 127 Then, as 127: 26: 33, the proportional part. 532627, the first log. Gives, with the index, 1.532660 for the log. of 34.0926. 4. If the number consist both of integers and fractions, or is entirely fractional; find the decimal part of the logarithm the same as if all its figures were integral; then this, having prefixed to it the proper index, will give the logarithm réquired. 5. And if the given number be a proper vulgar fraction: subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought; which, being that of a decimal fraction, must always have a negative index. 6. But if it be a mixed number; reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before. II. TO FIND THE NATURAL NUMBER TO ANY GIVEN LOGARITHM. THIS is to be found in the tables by the reverse method to the former, namely, by searching for the proposed logarithm among those in the table, and taking out the corresponding number by inspection, in which the proper number of integers are to be pointed off, viz. 1 more than the index. For, in finding the number answering to any given logarithm, the index always shows how far the first figure must must be removed from the place of units, viz. to the left hand, or integers, when the index is affirmative; but to the right hand, or decimals, when it is negative. EXAMPLES. So, the number to the log. 1-532882 is 34.11. But if the logarithm cannot be exactly found in the table; take out the next greater and the next less, subtracting the one of these logarithms from the other, as also their natural numbers the one from the other, and the less logarithm from the logarithm proposed. Then say, As the difference of the first or tabular logarithms, Is to the difference of their natural numbers, So is the differ. of the given log. and the least tabular log. To their corresponding numeral difference. Which being annexed to the least natural number above taken, gives the natural number sought, corresponding to the proposed logarithm. EXAMPLE. So, to find the natural number answering to the given logarithm 1.532708. Here the next greater and next less tabular logarithms, with their corresponding numbers, are as below: Next greater 532754 its num. 341000; given log. 532708 Next less 532627 its num. 340900; next less 532627 Differences 127 100 81 Then, as 127: 100 :: 81 : 64 nearly, the numeral differ. Therefore 34-0964 is the number sought, marking off two integers, because the index of the given logarithm is 1. Had the index been negative, thus sponding number would have been cimal. 1.532708, its corre340964, wholly de MULTIPLI 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 table, take out the natural number, answering to the sum, for the product sought, Observing to add what is to be carried from the decimal part of the logarithm to the affirmative index or indices, or else subtract it from the negative. Also, adding the indices together when they are of the same kind, both affirmative or both negative; but subtract ing the less from the greater, when the one is affirmative and the other negative, and prefixing the sign of the greater to the remainder. 3. To mult. 3.902 and 597·16 | 4.To mult.3 586, and 2.1046, Here the 2 cancels the 2, Prod. 0.1857618-1268956 and the 1 to carry from the decimals is set down. Here the 2 to carry cancels the-2, and there remains the -1 to set down. DIVISION 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. Observing to change the sign of the index of the divisor, from affirmative to negative, or from negative to affirmative; then take the sum of the indices if they be of the same name, or their difference when of different signs, with the sign of the greater, for the index to the logarithm of the quotient. And also, when 1 is borrowed, in the left-hand place of the decimal part of the logarithm, add it to the index of the divisor when that index is affirmative, but subtract it when negative; then let the sign of the index arising from hence be changed, and worked with as before. EXAMPLES. 1. To divide 24163 by 4567. 2. To divide 37-149by 523.76. Numbers. Logs. Dividend 24163 4.383151 Numbers. Logs. Dividend 37.149 1.569947 0.723520 Quot. '0709275 -2.850815 Divisor 4567 3.659631 Quot. 5.29078 Note. As to the Rule-of-Three, or Rule of Proportion, it is performed by adding the logarithms of the 2d and 3d terms, and subtracting that of the first term from their sum. INVOLUTION 1 |