What is the characteristic? The number has five figures therefore the characteristic is 4 and the log of 37374 equals 4.5725696 A slight further alteration in the method of finding the logs of numbers containing more than five figures must now be noticed. Most tables do not give the logs of numbers higher than 108000. (d) What is the log of 46176532? Look out the mantissa of the log of the first five figures 46176. The mantissa of the log of 46176 = 6644163. Now refer to the extreme right-hand column headed Diff.' Choose the nearest small column, in this case the only one, headed 94. By means of the figures in this column we are enabled to ascertain what amount must be added to the log of 46176 in respect of the remaining figures 532, in the following way : The positions of the difference figures should be carefully noticed. (e) What is the log. of 6644232 ? ... Log of 66442326-82244483 (f) What is the log of 105557 ? 10555 = 0234582 288 If the log of 105557 is looked out on page 197 it will be found to one more place of decimals, viz. 5.02348704. Pages xi and xii, sec. 6 d, show a slightly different method of finding the logs of numbers containing more than five figures without recourse to the Diff.' column. · Some of the figures in the tables have a line drawn over them. This means that the four figures over which the line is drawn are to be preceded by the three figures given just after instead of before them. Example:-The log of 82604 = 4.9170011 and not 4-9160011. Also the number corresponding to the log 4.9180041 and not 82805. 82795 We have considered the looking out of the log of a number, we must now consider the looking out of the number which corresponds to the log. The process will, of course, be exactly the reverse. To find the number which corresponds to a given log.-Look in the tables for the nearest log just below the given one. The difference between this and the given log must be used for finding out what figures must be added to the five figures of the number already found. Consult the small difference column at the side and take the nearest difference figure given in it, which is just below the difference of the logs already found. Against this will be found the sixth figure of the number. Subtract, add 0 to the remainder, and again look in the 'Diff.' column for the next below and so obtain the seventh figure of the number. Repeat until sufficient figures are obtained. A few examples should make the above explanation quite clear. (a) What is the number whose log is 4.3678921? Look in the tables for the log next below and subtract it from 4.3678921. Then look in the diff.' column for the nearest 'diff.' number just below. NOTE.-As this process is the reverse of finding the log, the numbers must contain one more figure to the left of the decimal point than is indicated by the characteristic of the given log. In the example just given the characteristic of the given log is 4, therefore the number must contain 5 figures to the left of the decimal point as shown. (b) What is the number whose log is 6·1234321 ? Further decimal places could be found if required. As the characteristic of the given log is 6, therefore the number must contain 7 figures to the left of the decimal point as shown. If the given log contain more than seven places of decimals those beyond the seventh must be brought down instead of adding O's. (c) What is the number whose log is 3.987654222 ? It is as well to finish off, as we have done, with the nearest difference whether above or below. ... The number 9719-7293. THE USE OF LOGARITHMS To multiply two numbers together. Find the log of each number. Add them together and the total will be the log of the To divide one number by another.-Find the log of each number, subtract the log of the divisor from the log of the dividend, and the result will be the log of the answer. Example.-Divide 0.06321 by 124.56. Log. of 0.06321 2.8007858 Log of 124.56 2.0953786 = It will be noticed in the above example that calculations with negative characteristics are made according to the rules of algebraic subtraction. The sign of the bottom number is changed and added to the top. Thus to subtract 2 from 2 the result is found by adding 2 to 2 which gives 4. To perform involution (i.e. to raise a number to a required power). -Find the log of the number; multiply it by the exponent of the power and the result will be the log of the answer. Example.-(a) Find the square of 731.77. To perform evolution (i.e. extract the root).-Divide the logarithm of the given number by the exponent of the root, and the result will be the log of the answer. Example. (a) Find the cube root of 54321. (b) Find the cube root of 0.054321 Log of 0.054321 = 2.7349678 Here the negative characteristic is not divisible by the divisor 3, so that we must add such a negative number to it as will make it divisible, and prefix an equal positive integer to the fractional part of the log. See pages xiii, xiv, xv, and xvi in the Tables. |