2.763353, By observing these logarithms, it will be seen, that THE NEGATIVE CHARACTERISTIC OF A LOGARITHM, SHOWS HOW FAR THE FIRST SIGNIFICANT FIGURE OF ITS CORRESPONDING DECIMAL IS DIS TANT FROM THE UNITS' PLACE. EXPLANATION OF THE TABLE. When the logarithms of numbers, from 1 upward to any other given number, are calculated, and arranged in a table, they constitute a table of logarithms. Tables of logarithms of great extent have been calculated, but those in common use extend to numbers no higher than 10,000. This is far enough for the purposes of ordinary calculation. In the first column on the left of each page, are arranged the natural numbers, and to distinguish this column from the others, the letter N is placed over it. Opposite these numbers in the next ten columns, are arranged the logarithms. For the first 100 numbers, however, on the first page of the table, there is but a single row of logarithms, opposite each column of numbers, and there are four rows of numbers on the page. TO FIND THE LOGARITHM OF ANY WHOLE NUMBER. If the number be less than 100, look for it in the column headed N, and directly opposite to it, in the column headed Log. will be found its logarithm. If the number be greater than 100, but less than 1,000, find it as before, in the column headed N, and directly opposite, in the column headed 0, will be found its logarithm. It will be seen, that when the first two figures of several successive logarithms are alike, they are omitted in the table, after having been once inserted, and only four figures are retained. When, therefore, there are but four figures in the logarithm opposite the given number, cast the eye up the blank till you find two more, which prefix to those already found. It will likewise be observed, that for the logarithms of numbers above 100, no characteristic is inserted in the table. But this may easily be supplied, since it is always a unit less than the integral places in the number. For numbers between 100 and 1,000, then, it is 2. Find the logarithm of 863. Opposite this number in the table, are found the figures 8520. Casting the eye up the blank we meet with 93, which we prefix to 8520, making 938520. The characteristic, 2, being then prefixed, we have the complete logarithm, 2.938520. If the number be greater than 1,000, and less than 10,000, find the first three figures in the column, headed N, and the fourth at the head of the page. Then, exactly opposite the first three, and in the column headed by the fourth, will be found the last four figures of the required logaTo these must be prefixed two others, found as above, in the column headed 0. rithm. contain six figures, the first two of those must be prefixed. If not, cast the eye up the blank, in search of the proper ones, as before. There is one exception to this rule. In some logarithms, will be seen points or periods occupying the places of figures. When this is the case, ciphers must be written in place of the points, and the two figures to be prefixed, must be sought in the next lower line, in the column, headed 0. Thus, Find the log. of 4,177. Opposite 417, and in the column headed 7, are found the figures 0864. The opposite log. in the column headed 0, has six figures, of which the first two, viz. 62, are to be prefixed to 0864, making 620864. The characteristic being joined, the log. is 3.620864. For 4,143, we find opposite 414, and under 3, the figures 7315. Here we must look along up the blank in the column, headed 0, for the two figures to be prefixed, which are 61, making the log. 617315, which, with its characteristic, is 3.617315. For 4,366, we find opposite 436, and under 6, the figures..84. Here, therefore, we must write ciphers for the periods, thus, 0084, and look in the next lower line of the column headed 0, for the two figures to be prefixed; which are 64, making the log. 640084, which, with its characteristic, is 3.640084, If the number exceed 10,000, find the logarithm of the first four figures as above. By the remaining figures, multiply the number opposite, in the column headed D; from the right of the product, reject as many figures as the given number has places more than four, and add what is left to the logarithm previously found. This sum will be the log. required. The column, marked D, is a column of differences: that is, it contains the differences between each two successive logarithms. In many cases a mental estimate may be made, of what should be added to a log. without the use of this column. The logarithms of pure or mixed decimal numbers should be taken out as if for whole numbers, care being taken to prefix the proper characteristic, in each case. Of vulgar fractions the logs. may be found by reducing them first to decimals, or by the rule given below, in division. TO FIND THE NUMBER CORRESPONDING TO ANY GIVEN LOGARITHM. Find the logarithm next less than that given, in the column headed 0; pass the eye to the right, along that horizontal line, into the other columns, and you will find either the log. given, or one very near it; the first three figures of the corresponding number will then be found opposite, in the column headed N, and the fourth figure, directly over the log. at the top of the page. The integral places in this number will be determined by the characteristic of the log. If this characteristic be 3, all the places will be integral; if it be 2, one decimal must be pointed off; if it be 1, two decimals, &c. On the other hand, if it be 4, a cipher must be annexed to the number; if it be 6, two ciphers must be annexed, and so on. When the given log. cannot be found in the table, however, the number may be more accurately obtained by the following process. Find in the table the next less log., and take the difference between it and the given one. Make this difference the numerator of a fraction, and the number opposite in the column headed D, the denominator; reduce this fraction to a decimal, which write immediately after the four figures corresponding to the tabular log. used. Afterwards place the decimal point where the characteristic of the given log. may require. lar log. is .954918, and the difference 3. Making this difference the numerator, and 48, the opposite number in the column headed D, the denominator of a fraction, we have 3 This, reduced to a decimal, gives .0625, which, annexed to 9014, the number corresponding to .954918, makes 90140625. Pointing off according to the given characteristic, we have 901.40625, the number required. Allowance may often be made in the mind, without the trouble of calculation. MULTIPLICATION BY LOGARITHMS. From the nature of logarithms, as above explained, the reason of the following rule will be obvious. TAKE FROM THE TABLE THE LOGARITHMS OF THE NUMBERS TO BE MULTIPLIED, AND ADD THEM TOGETHER. ARITHM OF THE PRODUCT. THEIR SUM WILL BE THE LOG EXAMPLES. 1. Multiply 29.15 by 9.4635. Log. 29.15 1.464639 Log. 9.4635-0.976052 Product, 275.86139+2.440691, sum. When there are negative characteristics, it is best to neglect them entirely in the addition, and afterwards diminish the resulting characteristic by their amount. Thus, 2. Multiply 71.32 by .3205. Log. 71.32-1.853211 Sum, neglecting T=2.359039 INVOLUTION BY LOGARITHMS. A number is involved by multiplying it one or more times into itself. But a number is multiplied by itself, by adding its log. to itself; that is, a number is raised to the second power, by multiplying its log. by 2. In like manner, a number is raised to the third power, by multiplying its log. by 3, and so on. Hence, to perform involution by logarithms, MULTIPLY THE LOGARITHM OF THE NUMBER TO BE INVOLVED, BY THE INDEX OF THE POWER. THE PRODUCT WILL BE THE LOGARITHM OF THE Log. power 4T.294089. Therefore the power is .000000000000000000000000000000000000000019683. Here there were 4 to carry, which made the negative product, 45, equal to TT. DIVISION BY LOGARITHMS. To divide one number by another, SUBTRACT THE LOGARITHM OF THE DIVISOR FROM THAT OF THE DIVITHE REMAINDER WILL BE THE LOGARITHM OF THE QUOTIENT. DEND. EXAMPLES. 1. Divide 15,625 by 625. Log. 15,625 4.193820 Log. 625-2.795880, Quotient, 25-1.397940 In case the characteristic of the divisor is negative, it must be added, instead of being subtracted. Thus, 2. Divide 339 by 0.0807. Log. 339-2.530200 Quotient, 4,201.25 nearly=3.623326 In case the characteristic of the dividend is negative, or is less than that of the divisor, it is best to increase it by a number, which will make it so great, that the divisor may be taken from it. After the process of subtraction, the same number should be taken away from the resulting characteristic. The best number for this purpose is 10, or some even number of tens. Thus, 3. Divide 77 by 111. Log. 77-1.886491; increased by 10-11.886491 Difference, Quotient, .6937= 2.045323 9.841168 T.841168 This gives us the means of finding the log. of a vulgar fraction; since a fraction merely indicates the division of its numerator by its denominator. The rule seems to be : TAKE THE LOG. OF THE DENOMINATOR FROM THAT OF THE NUMERATOR THE REMAINDER IS THE LOG. OF THE FRACTION. 4. Find the log. of 25. Log. 25-1.397940; increased by 10-11.397940 Difference, 1.672098 9.725842 EVOLUTION BY LOGARITHMS. As evolution is the opposite of involution, its rule must of course be, DIVIDE THE LOG. OF THE GIVEN NUMBER, BY THE NUMBER EXPRESSING THE ROOT TO BE Found. EXAMPLE. Find the 10th root of 59,049. Log. 59,049-4.771213 Divide by 10-0.477121 In the extraction of roots, logarithms will often be found extremely useful. PROPORTION BY LOGARITHMS. ADD TOGETHER THE LOGS. OF THE SECOND AND THIRD TERMS, AND FROM THE SUM SUBTRACT THE LOG. OF THE FIRST. THE REMAINDER WILL BE THF LOG. OF THE FOURTH TERM REQUIRED. EXAMPLE. If 23 lbs. of sugar cost $2.76, what cost 250 lbs.? EXPLANATION OF THE TABLE OF LOGARITHMIC SINES AND TANGENTS. This table consists merely of logarithms, calculated for the numbers which express the lengths of the natural sines and tangents, to every minute of a quadrant, whose radius is 10,000,000,000. The logarithm of RADIUS, of the sine of 90°, and of the tangent of 45°, will, of course, be 10.000000. As there is frequent necessity to employ the numbers in this table, the following rules must be thoroughly understood. TO FIND THE LOGARITHMIC SINE, CO-SINE, TANGENT, OR CO-TANGENT OF ANY NUMBER OF DEGREES, OR DEGREES AND MINUTES. If the degrees are less than 45, look for them at the top of the page, and for the minutes, in the left hand column; under the given name, and opposite the minutes, will be found the logarithm sought. But, if the degrees are between 45 and 90, look for them at the bottom of the page, and for the minutes, in the right hand column: over the given name, and opposite the minutes, will be found the logarithm sought. If the angle be more than 90°, find the log. for its supplement, and it will be |
