213. What is the greatest load that should be lifted with a pair of tackle blocks having three pulleys in the movable block and two in the fixed block, and using a 2-in. rope? FIG. 94.-Illustration for Problem 211. 100 lbs. per sq. in. CHAPTER XIX LOGARITHMS 131. Common Logarithms.-Logarithm is the name of a group of numbers the properties of which enable us to divide or multiply large numbers, or obtain powers and roots of numbers, more quickly and easily than by ordinary arithmetical processes. The Common System of logarithms, having the number 10 as its base, is the system in general use, and the one which we shall consider in this chapter. Any number has a corresponding logarithm, and this logarithm is fixed for that particular number. For example, the logarithm of 3 is 0.4771 and always has that value in this system. Some other logarithms are: Logarithm of 17 is 1.2304 Logarithm of 983 is 2.9926 132. Mantissas and Characteristics. Every logarithm is made up of two distinct parts, the part to the right of the decimal point called the mantissa, and the part to the left of the decimal point called the characteristic. In the logarithm of 3, mentioned above, the mantissa is .4771, and the characteristic is 0. The mantissa is determined by the significant figures in the number, without regard to the decimal point, and the characteristic is determined by the location of the decimal point in the number. To obtain the characteristic, the following simple method is used. If there is one figure to the left of the decimal point in the number, the characteristic of its logarithm is 0; if there are two figures, the characteristic is 1; three figures, the characteristic is 2; etc. In other words, the characteristic is always one less than the number of figures to the left of the decimal point in the number. To illustrate this rule, let us find the logarithms of the numbers, 3, 30, 300, and 3000. The mantissa for all four logarithms is .4771, since the significant figure without reference to the decimal point is 3 in all the numbers. Applying the above rule for the characteristic, it is evident that the characteristic for the log of 3 is 0; of 30 is 1; of 300 is 2; and of 3000 is 3. Now combining the characteristics and mantissas, we get 133. Use of Logarithmic Tables. A table of four-place logarithms for numbers from 1 to 1000 is given on pages 228 and 229. Larger tables are available in various publications and textbooks. The tables give only the mantissas of the logarithms. The characteristics must be determined by inspection. As stated before, the mantissas depend solely upon the significant figures in the number without any regard for the decimal point. Figures 95 and 96, showing sections of the logarithmic tables, demonstrate clearly the method of finding the mantissas. The significant figures are "12." Figure 95 illustrates how to find the mantissa from the tables. At the left-hand edge of the table are the numbers. The number 12 is indicated in heavy type. Its mantissa is in the first column to the right and is .0792 as indicated. The number 120 has three figures to the left of the decimal point, so the characteristic is 2. Therefore, Log 120 = 2.0792 Referring to the tables, it will be noted that the number column at the left-hand edge contains only numbers of two sig nificant figures. The mantissas for numbers of three significant figures are found as illustrated in the following example. Example: Find the logs of the following numbers: 6, 56.3, and 5870. In Fig. 96 the number 6 (or 60) is indicated in heavy type, and its mantissa, .7782, is in the first column to the right of it. The next number, 56.3, has three significant figures. The first two figures, 56, we locate in the lefthand column and the third figure, 3, in the row of figures from 0 to 9 at the top. The mantissa is found in the column beneath the 3 and opposite the 56, and its value .7505, as indicated. Similarly the mantissa, .7686, for the number 5870 is opposite the 58 and in the column beneath the 7. the mantissas we can write the logarithms. Knowing Note.-Work the following problems before proceeding with the rest of the chapter. 214. Find the logarithm of 13. 215. Find the logarithms of 6 and 9. 216. Find the logarithm of 82. 217. Find the logarithm of 825. 218. Find the logarithm of 126. 134. Interpolation. Since the tables shown do not contain numbers of more than three significant figures, it is necessary to interpolate to find the logs of numbers containing four or more significant figures. For example, we will find the log of 2364. The nearest numbers to this value for which we can find the man |