0 10 tissas are 2370 and 2360, and their mantissas are .3747 and .3729. The mantissa for 2364 is somewhere between these two values. The difference between the numbers 2370 and 2360 is 10. The number 2364 is 4 greater than 2360; thus 2364 is of the difference, or of 10 greater than 2360. Similarly, the mantissa for the log of 2364 is greater than .3729 (the mantissa of 2360) by 10 of the difference between their mantissas, .3747 and .3729. The difference between .3747 and .3729 is .0018, and of .0018 = .00072. Adding this correction to .3729 gives us the mantissa for the log of 2364. = To facilitate interpolation, a small table of corrections, or proportional parts, already worked out, is given at the right of the mantissas. By the use of these tables we can find the corrections very quickly. Let us take the example just worked-to find the log of 152,600. The first three figures are 152. The mantissa for 152 is opposite the "15" and below the "2." As found before, its value is .1818. The last significant figure in the number is 6, which we locate in the top row of figures in the correction table. Directly beneath it and opposite the mantissa, .1818, is the correction, 17, to be added to the mantissa. The new mantissa will then be .1818+ 17.1835. By inspection, the characteristic is 5, so the log of 152,600 is 5.1835. In the case of a number having more than four significant figures, it is sufficiently accurate in this work to drop the additional figures, giving the fourth figure the nearest integral values. Thus, if we were to find the log of 964.83, we would drop the 3, leaving the four significant figures 964.8+, and find the log as before. Or if the number were 964.862, we would drop the 62 and increase the fourth figure by one, making it 964.9. The student should check the following logarithms by the method just explained to be sure he understands it correctly. 135. Antilogarithms. When the logarithm of a number is known and we wish to find the number, we call it finding the antilogarithm. Suppose we have given the logarithm 3.4829. To find the antilogarithm is simply to find the number whose logarithm is 3.4829. By searching in the mantissa columns of the table, we find the mantissa .4829. It is found opposite the number 30 and in the column beneath the "4," so that the significant figures of the number are 304. The characteristic of 3 indicates four figures to the left of the decimal point. Pointing off four places we get the number, 3040, whose logarithm is 3.4829. If the exact mantissa does not appear in the tables, find the mantissa which has the nearest value less than the given mantissa. This determines the first three figures of the number. The fourth figure is found by taking the difference between the two mantissas and locating this difference (or the nearest value thereto) in the correction table on a horizontal line with the mantissa. The small number at the top, vertically above the correction, is the desired fourth figure of the number. Example: Find the antilog of 1.4836. The mantissa .4836 is not given in the table, and the nearest mantissa less than .4836 is .4829. The number corresponding to the mantissa .4829 is 304. 4836 4829 7 (difference). = Looking in the correction table, opposite 4829 is the figure 7. The corresponding number above the 7 is 5. Therefore, antilog 1.4836 30.45. = 136. Multiplication by Logarithms. To multiply by means of logarithms, add together the logarithms of the numbers to be multiplied, and the resulting sum is the logarithm of the product. The product is then obtained by finding the antilog of the sum. 137. Division by Logarithms.-Division is the reverse of multiplication, and so in division by logarithms the log of the divisor is subtracted from the log of the dividend to obtain the log of the quotient. (Multiply the numbers in the denominator and numerator separately and then divide.) 221. Find the antilog of 2.9119. 222. Multiply 63 X 180 X 4.3. 223. Divide 725 by 72. 138. Roots and Powers.-The calculation of roots and powers of numbers is simplified considerably by the use of logarithms. To obtain any root of a number, simply divide the log of the number by the figure which indicates the root. The quotient thus obtained is the log of the root desired, and its antilog is the root. To obtain any desired power of a number, multiply the log of the number by the figure which indicates the power. The product is the log of the desired power, and its antilog is the power. 139. Logarithms of Decimals.-Previous to this we have not considered the logarithms of numbers whose values are less than 1; in other words, decimal fractions. The logarithms of decimals, such as .03, .0024, etc., are somewhat more complicated since the characteristic becomes less than zero or what is called a negative number. A negative number is that obtained by subtracting any number from another smaller than itself, as, for instance, 4 - 6 = −2. The negative result is read as "minus 2." The mantissas for the logs of decimals are obtained exactly as they would be for any other number, since the decimal point does not affect the mantissa. The mantissa always has a positive value. As stated before, the characteristic is negative, and its value is one more than the number of ciphers, or zeros, immediately following the decimal point, that is, between the decimal point and the first significant figure. Thus, the mantissa for the log of .03 is .4771 and the characteristic is -2 (minus 2) since there is one cipher immediately following the decimal point. The log of .03 would thus be -2.4771. This method of writing the characteristic makes it rather difficult to use, so it is customary to substitute for the negative characteristic an equivalent subtraction. The mantissa and the characteristic are then combined as follows: Characteristic for log .03 = 2, or 8 10 The following logs will illustrate the point: = 7.8451 - 10 (characteristic is -3) Log .007 = = 9.8028 10 (characteristic is 1) Log .00041 6.6128 10 (characteristic is -4) = To make clear how these are used in multiplication, division, etc. we will work the following examples. The characteristic 18 = 20: -2, which is the same as 8 10, and indicates that there should be one cipher after the decimal point. The mantissa is .8252, and we find the number as explained in Art. 132. Antilog 18.8252 - 20 = .06687, Answer. To divide we must subtract the log of .00872 from the log of 825, which it is evident we cannot do in the form they are now written. However, by |