LOGARITHMS 331. Early in the seventeenth century, John Napier, a Scotchman, invented logarithms, by the use of which the arithmetical processes of multiplication, division, evolution and involution are greatly abridged. 1/0 4 2 83 16 4 32 5 64 6 128 7 332. Many simple arithmetical operations can be performed by the use of two columns of numbers, as given in the annexed table. The left-hand column is formed by writing unity at the top and doubling each number to get the next. The right-hand column is formed by writing opposite each power of 2, the index of the power. Thus 512 = 29, the number opposite 512 indicating the power of 2 used to produce 512. Ex. 1. Multiply 4096 by 64. 256 8 512 9 1024 10 2048 11 4096 12 8192 13 131072 17 The student should notice that the simple operation of addition is substituted for multiplication, the product being found in the left-hand column opposite 18, the 1048576 20 sum of 12 and 6. 262144 18 524288 19 Ex. 2. Divide 1048576 by 2048. 1048576 2048 220 211 = 220-11 = place of division). 29 512 (subtraction takes the In the preceding table the numbers in the right-hand column are called the logarithms of the corresponding numbers in the left-hand column. 2 is called the base of this system. Therefore, the logarithm of a number is the exponent by which the base is affected to produce the number. 333. Any other base than 2 might have been used and columns similar to the above formed. In practice 10 is always taken as the base and the logarithms are called common logarithms in distinction from the natural logarithm, of which the base is 2.71828. Common logarithms are indices, positive or negative, of the power of 10. From the definition of common logarithms, it follows that since = 334. Since most numbers are not exact powers of 10, logarithms will in general consist of an integral and decimal part. Thus, since log 100 2 and log 1000 = 3, the logarithms of numbers between 100 and 1000 will lie between 2 and 3, or will be 2+ a fraction. Also since log 0.012 and log 0.001 = -3, the logarithms of all numbers between 0.01 and 0.001 will lie between 2 and - 3 or will be 3+ a fraction. The integral part of the logarithm is called the characteristic and the decimal part the mantissa. 335. The characteristic of the logarithm of a number is independent of the digits composing the number, but depends on the position of the decimal point. Characteristics, therefore, are not given in the tables. Thus, since 246 lies between 100 and 1000, log 246 will lie between 2 and 3, or will be 2+ a fraction. Again since 0.0024 lies between 0.001 and 0.01, its logarithm lies between - 3 and - 2, or log 0.0024 = − 3 + a fraction. 336. From the above illustrations it readily appears that the characteristic of the logarithm of a number, partly or wholly integral, is zero or positive and one less than the number of figures in the integral part. 337. The characteristic of the logarithm of a pure decimal is negative and one more than the number of zeros preceding the first significant figure. EXERCISE 77 1. Determine the characteristic of the logarithm of 2; 526; 75.34; 0.0005; 300.002; 0.05743. 2. If log 787 = 2.8960, what are the logarithms of 7.87, 0.0787, 78700, 78.7? 338. The mantissa of the logarithm of a number is independent of the position of the decimal point, but depends on the digits composing the number. Mantissas are always positive and are found in the tables, for moving the decimal point is equivalent to multiplying the number by some integral power of 10, and therefore adds to or subtracts from the logarithm an integer. Thus, log 76.42 = log 76.42, log 764.2 = log 76.42 × 10 = log 76.42 + 1, log_7642 = log 76.42 × 102 = log 76.42 + 2, = log 7.642 = log 76.42 x 10-1 log 76.42 +(−1). So that the mantissas of all numbers composed of the digits 7642 in that order are the same, since moving the decimal point affects the characteristic alone. Log 0.0063 is never written - 37993, but 3.7993. The minus sign is written above to indicate that the characteristic alone is negative. To avoid negative characteristics 10 is added and subtracted. Thus, 3:7993 = 7.7993 - 10. 339. The principles used in working with logarithms are as follows: I. The logarithm of a product equals the sum of the logarithms of the factors. II. The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. III. The logarithm of a power equals the index of the power times the logarithm of the number. IV. The logarithm of a root equals the logarithm of the number divided by the index of the root. Given log 2=0.3010, log 3=0.4771, log 5=0.6990, find: 9. Find the number of digits in 3025; in 2530. USE OF TABLES 340. In the tables here given the mantissas are found correct to but four decimal places. By using these tables results can generally be relied upon as correct to 3 figures and usually to 4. If a greater degree of accuracy is required, five-place or even seven-place tables must be used. 341. To find the logarithm of a given number. Write the characteristic before looking in the tables for the mantissa. Find the mantissa in the tables. (1) When the number consists of not more than three figures. In the column N, at the left-hand side of the page, find the first two figures of the number. In the row N, |