By putting = 2x and y = x in (11), (12), and (15), 28. sin (2x + y) − 2 sin x cos (x + y) = sin y. 31. 1+tan x tan 2x = sec 2x. 32. sin 4x = 4 sin x cos x 8 sin3x cos x. 33. cos 4 x 8 cos1x-8 cos2x+1. 34. sin 5x=5sinx − 20 sinx +16 sinx. 35. By putting x 45° and y = 30° in (13) and (14), Art. 36. By putting x = 30° in (34) and (35), Art. 75, prove 38. By putting x = 45° in (34) and (35), Art. 75, prove tan 22° 30′ = √2 — 1, cot 22° 30' = √√2 + 1. VI. LOGARITHMS. 79. Every positive number may be expressed, exactly or approximately, as a power of 10; thus, When thus expressed, the corresponding exponent is called its Logarithm to the base 10; thus, 2 is the logarithm of 100 to the base 10, a relation which is written = log10 100 2, or simply log 100 = 2. And in general, if 10o : =m, then x = log m. 80. Any positive number except unity may be taken as the base of a system of logarithms; thus, if a*=m, then x = logam. Logarithms to the base 10 are called Common Logarithms, and are the only ones used in numerical computation. If no base is expressed, the base 10 is understood. Note. The second form of the results for log.1, log .01, etc., is preferable in practice. 82. It is evident from Art. 81 that the logarithm of a number greater than 1 is positive, and that the logarithm of a number between 0 and 1 is negative. 83. If a number is not an exact power of 10, its common logarithm can only be expressed approximately; the integral part of the logarithm is called the characteristic, and the decimal part the mantissa. For example, log 13 = 1.1139. In this case the characteristic is 1, and the mantissa .1139. 84. It is evident from the first column of Art. 81 that the logarithm of any number between 1 and 10 is equal to 0 plus a decimal ; 10 and 100 is equal to 1 plus a decimal; 100 and 1000 is equal to 2 plus a decimal; etc. Hence, the characteristic of the logarithm of a number with one figure to the left of its decimal point, is 0; with two figures to the left of the decimal point, is 1; with three figures to the left of the decimal point, is 2; etc. 85. In like manner, from the second column of Art. 81, the logarithm of a decimal between 1 and .1 is equal to 9 plus a decimal — 10; .1 and .01 is equal to 8 plus a decimal — 10; .01 and .001 is equal to 7 plus a decimal — 10; etc. Hence, the characteristic of the logarithm of a decimal with no ciphers between its decimal point and first significant figure, is 9, with -10 after the mantissa; of a decimal with one cipher between its point and first figure, is 8, with -10 after the mantissa; of a decimal with two ciphers between its point and first figure, is 7, with -10 after the mantissa; etc. 86. For reasons which will be given hereafter, only the mantissa of the logarithm is given in tables of logarithms of numbers; the characteristic must be supplied by the reader. The rules for characteristic are based on Arts. 84 and 85: I. If the number is greater than 1, the characteristic is 1 less than the number of places to the left of the decimal point. II. If the number is between 0 and 1, subtract the number of ciphers between the decimal point and first significant figure from 9, writing -10 after the mantissa. Thus, characteristic of log 906328.5 = 5; characteristic of log.007023 = = 7, with - 10 after the mantissa. Note. Some writers, in dealing with the characteristics of negative logarithms, combine the two portions of the characteristic, writing the result as a negative characteristic before the mantissa. Thus, instead of 7.6036 — 10, the student will frequently find 3.6036, a minus sign being written over the characteristic to denote that it alone is negative, the mantissa being always positive. PROPERTIES OF LOGARITHMS. 87. In any system, the logarithm of unity is zero. For since a = 1, we have log. 1 = 0 (Art. 79). 88. In any system, the logarithm of the base itself is unity. For since a1 = a, we have log, a = 1. 89. In any system whose base is greater than unity, the logarithm of zero is minus infinity. 90. In any system, the logarithm of a product is equal to the sum of the logarithms of its factors. Multiplying, we have Whence, a2 × a3 = mn, or a*+y = mn. loga mn = x+y. Substituting the values of x and y, we have log mn = log m+log, n. In like manner, the theorem may be proved for the product of three or more factors. 91. By aid of the theorem of Art. 90, the logarithm of any composite number may be found when the logarithms of its factors are known. 1. Given log 2 = .3010, log 3 = .4771; find log 72. log 72 = log (2 × 2×2×3×3) = log 2+ log 2+ log 2 + log 3 + log 3 = 3 × log 2 + 2 × log 3 = .9030.9542 = 1.8572. EXAMPLES. Given log 2.3010, log 3.4771, log 5.6990, log 7 = .8451; find the values of the following: 92. In any system, the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. |