INTRODUCTION. LOGARITHMS. 1. Definition. A logarithm of a number is the exponent denoting the power to which a fixed number, called the base, must be raised in order to produce the given number. Thus, in the equation, bn, b is the base of the system, n is the number whose logarithm is to be taken, and x is the logarithm of n to the base b, which may be written: x-log, n. Any positive number, except 1, may be assumed as the base, but when assumed, it remains fixed for a system; hence, there may be an infinite number of systems, since there may be an infinite number of bases. 2. Common Logarithms. Common logarithms are the logarithms of numbers in the system whose base is 10. Hence, In the common system, the logarithm of an exact power of 10 is the whole number equal to the exponent of the power. (9) 3. Consequences. 1. If the number is greater than 1 and less than 10, its logarithm is greater than 0 and less than 1, or is 0 + a decimal 2. If the number is greater than 10 and less than 100, its logarithm is greater than 1 and less than 2, or is 1 a decimal. 3. In general, if the number is not an exact power of 10, its logarithm, in the common system, will consist of two parts—an entire part and a decimal part. The entire part is called the characteristic and the decimal part is called the mantis 4. Problem. To find the laws for the characteristic. Let (1) 10*= n; then, by def., log n = x. 10. (2) = 66 = 66 (3) 10-1 Log 89793.95323. 66 897.9 = 2.95323. 89.79 = 1.95323. 8.979 0.95323. = n 16; then, by def., log 10 n ... log log n-1. 10 Hence, The logarithm of the quotient of any number by 10 is less by 1 than the logarithm of the number. = Let us now take the number 8979 and its logarithm 3.95323, as given in a table of logarithms, and divide the number successively by 10, and for each division subtract 1 from the logarithm of the dividend, then we have, Log .8979 66 η 10 66 .08979 .008979 The minus sign applies only to the characteristic over which it is placed. The mantissa is always positive, and is the same for all positions of the decimal point. An inspection of the above will reveal the following laws: 1. If the number is integral or mixed, the characteristic is positive and is one less than the number of integral figures. 2. If the number is entirely decimal, the characteristic is negative and is one greater, numerically, than the number of O's immediately following the decimal point. 5. Exercises on the Characteristic. 1. What is the characteristic of the logarithm of 7? 2. What is the characteristic of the logarithm of 465? 3. What is the characteristic of the logarithm of 4678? 4. What is the characteristic of the logarithm of 34.75? 5. What is the characteristic of the logarithm of .65? 6. What is the characteristic of the logarithm of .0789? 7. What is the characteristic of the logarithm of .00084 ? 8. If the characteristic of the logarithm of a number is 2, how many integral places has that number? 9. If the characteristic of the logarithm of a number is 5, how many integral places has that number? 10. If the characteristic of the logarithm of a number is 1, how many integral places has that number? 11. If the characteristic of the logarithm of a number is 0, how many integral places has that number? 12. If the characteristic of the logarithm of a number is negative, is the number integral, decimal, or mixed? 13. If the characteristic of the logarithm of a number is 4, how many O's immediately follow the decimal point? 14. If the characteristic of the logarithm of a number is 2, how many O's immediately follow the decimal point? 15. If the characteristic of the logarithm of a number is 1, how many O's immediately follow the decimal point? TABLE OF LOGARITHMS. 6. Description of the Table. The table of logarithms annexed gives the mantissa of the logarithm of every number from 1000 to 10900. The characteristic can be found by the preceding laws. It follows, from Art. 4, that the mantissa of the logarithm of a number is the same as the mantissa of the logarithm of the product or quotient of that number by any power of 10. Thus: Hence, we can determine from the table the logarithm of any number less than 1000. Thus, the mantissa of the logarithm of 8 is the same as that of the logarithm of 8000. In the table, the first three or four figures of each number are given in the left-hand column, marked N. The next figure is given at the head and foot of one of the columns of ma tissas. The mantissas, in the column under 0, are given to five decimal places. The first and second decimal figures of this column are understood to be repeated in the spaces below, and to be prefixed, across the page, to the three figures of the remaining columns. When the third decimal digit changes from 9 to 0, the second is increased by the 1 carried; and the corresponding mantissa, and all to the right, commence with a smaller figure, to indicate that the first two decimal figures, to be prefixed, are to be taken from the line below. The last column, marked D, contains the difference of two successive mantissas, called the tabular difference. 7. Problem. To find the logarithm of a given number. 1. Find the logarithm of 3675. The characteristic is 3. Opposite 367, in the column. headed N, and under the column headed 5, we find 526, to which prefix the two figures, 56, in the column headed 0, and we have for the mantissa .56526. .'. log 3675 3.56526. 2. Find the logarithm of 76. The characteristic is 1, and the mantissa is the same as that of 7600, which is .88081. log 76 1.88081. 3. Find the logarithm of .004268. The characteristic is 3, and the mantissa is the same as that of 4268. Looking opposite 426, and under 8, we find 022, of which the 0 is a small figure. Prefixing |