column O is to be prefixed, and the characteristic, according to the Rule in Art. 242. In the columns 1, 2, 3, &c., a small cipher (0) or figure (1) is sometimes placed below the first figure, to show that the figure which is to be prefixed from the column O has changed to the next larger number, and is to be found in the horizontal line directly below. Thus, the log of 7960 is 3.9009 66 246. To find the logarithm of a number of more than three figures. On the right half of pages 268 and 269 are tables of Proportional Parts. The figures in any column of these tables are as many tenths of the average difference of the ten logarithms in the same horizontal line as is denoted by the number at the top of the column. The decimal point in these differences is placed as though the mantissas were integral. 1. To find the logarithm of a number of four figures, find as before the logarithm of the first three figures; to this, from the table of Proportional Parts, add the number standing on the same horizontal line and directly under the fourth figure of the given number. Thus, to find the log of 5743. The log of 5740 is 3.7589 In "Proportional Parts," in the same line, under 3, " 2.3 Therefore, the log of 5743" 3.7591 It is always best to find the logarithm of the nearest tabulated number, and add or subtract, as the case may be, the correction from the table of Proportional Parts. Whenever the fractional part omitted is larger than half the unit in the next place to the left, one is added to that figure. 2. For a fifth or sixth figure the correction is made in the same manner, only the point must be moved one place to the left for the fifth, two for the sixth, figure. Thus, to find the log of 3.6825. The logarithm of a common fraction may best be found by reducing the fraction to a decimal, and then proceeding as above. 247. To find the number corresponding to a given logarithm. Find, if possible, in the table the mantissa of the given logarithm. The three figures opposite in the column N, with the number at the head of the column in which the logarithm is found, affixed, and the decimal point so placed as to make the number of integral figures correspond to the characteristic of the given logarithm, as taught in Art. 242, will be the number required. Thus, The number corresponding to log 5.5378 is 345000 If the mantissa of the logarithm cannot be exactly found, take the number corresponding to the mantissa nearest the given mantissa; in the same horizontal line in the table of proportional parts find the figures which express the difference between this and the given mantissa; at the top of the page, in the same vertical column, is the correction that belongs one place to the right of the number already taken, to be added if the given mantissa is less, subtracted if greater. Thus, 1. To find the number corresponding to 2.7660 - 2.7657, and number corresponding, 583. log next less log, difference, 3 Number required, log 2. To find the number corresponding to 3.8052 next greater log, 3.8055, and number corresponding, 0.00639 difference, Number required, 3 correction, 44 0.0063856 The nearest number in the table of Proportional Parts to 3 is 2.7; corresponding to this at the top is 4, which belongs as a correction one place to the right of the number (0.00639) already taken, but 32.7 0.3; this, in like manner, gives a still further correction of 4, one place farther still to the right. The whole correction, therefore, is 44, to be deducted as shown in the operation above. = 3. Find the log of 3764. 4. Find the log of 2576000. 5. Find the log of 7.546. 6. Find the log of 0.0017. 3.5757 6.4109 0.8778 312304 7. Find the log of 1.3717 8. Find the number to log 3.807873. 6425.5 9. Find the number to log 1.820004. 10. Find the number to log 2.982197. 66.07 959. 11. Find the number to log 2.910037. 108129 12. Find the number to log 4.850054. 248. The great utility of logarithms in arithmetical operations is that addition takes the place of multiplication, and subtraction of division, multiplication of involution, and division of evolution. That is, to multiply numbers, we add their logarithms; to divide, we subtract the logarithm of the divisor from that of the dividend; to raise a number to any power, we multiply its logarithm by the exponent of that power; and to extract the root of any number, we divide its logarithm by the number expressing the root to be found. This is the same as multiplication and division of different powers of the same letter by each other, and involving and evolving powers of a single letter or quantity; the number 10 takes the place of the given letter, and the logarithms are the exponents of 10. MULTIPLICATION BY LOGARITHMS. RULE. 249. Add the logarithms of the factors, and the sum will be the logarithm of the product (Art. 50). 1. Multiply 347.6 by 0.04752. Ans. 16.517. 2. Find the product of 0.568, 0.7496, 0.0846, and 1.728. Ans. 0.06224. (It must be carefully borne in mind that the mantissa of the logarithm is always positive.) 4. Multiply 0.0004756 by 1355. Although negative quantities have no logarithms (Art. 262), yet, since the numerical product is the same whether the factors are positive or negative, we can use logarithms in multiplying when one or more of the factors are negative, taking care to prefix to the product the proper sign according to Art. 48. When a factor is negative, to the logarithm which is used n is appended. 5. Multiply -0.7546 by 54.5. 6. Find the product of -0.017, 25, and 165.4. 7. Find the product of -14, -7.643, and −0.004. Ans.-.428. DIVISION BY LOGARITHMS. RULE. 250. From the logarithm of the dividend subtract the logarithm of the divisor, and the remainder will be the logarithm of the quotient (Art. 54). |