46 EXPONENTIAL EQUATIONS. This process may be continued at pleasure, but we may now consider 3 as the approximate value of "". By substitution we shall find Let x=1+ 1 x'=2+ ·=2+· 3 X = 8 1 1 '' 2+ 1 -1+1, then we have then 2+ ($) + + 5 = in which is between 2 and 3. Let x'=2+ 27 Therefore is an approximate value of x; that is, 10%, or the 10 10th root of the 27th power of 10, is nearly equal to 500. 27 The exponent may also be expressed in the decimal form 2.7 10 and we may write =8, or 102.7-500. 55. As another example take the equation 8x=20, in which is evidently between 1 and 2. =1+ 32 3 7 in which x" is between 3 and 4. 8 { ગ 2 or 32 25 or 2 1 or = which equation will be more convenient if we employ decimals. Taking the decimals to two places, it is approximately (1.19)"'-1.28 in which "" is found to be between 1 and 2. then x'" =1+ = 78125 !!! 65536 x'=2+ =1+ 1.28, or (1.19) 32 25 x"=3+ =3+ = (1.08)xiv=1.19, in which xiv is found to be between 2 and 3. Taking 2 as its ap proximate value, we shall have 1+ = 5 '3 ·2+ 25 11 11 253 32. 11 36 25 25 The approximate value of x is therefore 36 25' 36 813—20, or 81.44—20 nearly. = 1.08, or or 1.44; whence That is, "the 25th root of the 36th power of 8 is nearly 20." As to the decimal form of the exponent 1.44, we may remark that 48 EXPONENTIAL EQUATIONS. 44 144 it is equal to 1 and may 100 100' 100th root of the 144th power." So also in the equation 102.778-600 or 2778 and the equation may be read, 1000' the exponent is equivalent to "the 1000th root of the 2778th power of 10 is equal to 600." If the decimal were taken to four places, we should supply the denominator 10000 or 104; if to five places, the denominator 105; and in general, a decimal fraction with n places of decimals, may be converted into a vulgar fraction by placing under it 10" and removing the decimal point. The decimal form is the most convenient when the exponent is not an exact fraction. 56. If in the equation a=b one of the quantities (a and b) is less than unity, and the other greater, the value of x will be negative. Thus, in the equation we find x=-3, for 2-3 In the equation (음)-2-3.375 8 In the equation 8x= whence 1 2 8-y 1 23 = 2x = 1 3.375, we find x=-3, for we find x 1 therefore be read as "the 20 Then since x is negative, let x=—y, and we have 1 1 1 or 8v=20, 1 3' = for 8-3 An equation of this kind is solved as follows. Take, for example, 1 8x 1 38 which may be solved as in the preceding article. We shall find y=1.44, and x=-y=-1.44. Therefore 1. Find the value of x in the equation 3o=15. 2 Find the value of x in the equation 10x=3. 4. Find the value of x in the equation 10~ CHAPTER VI. Ans. x 2.46. Ans. x 0.477. Ans. x 2.477. 1 =1. 3' Ans. -0.477. NATURE AND USE OF LOGARITHMS. 57. The nature of logarithms will be readily understood by considering the exponential equation ax=b. We have seen in the last chapter that if a and b are given, the value of x may be determined. In fact, a and b may be any two numbers whatever, and x will have a corresponding value which will satisfy the equation. If we suppose a to have a constant value as 10, and b to be successively 1, 2, 3, &c., we shall have the equations 10x=1, 10x=2, 10x=3, &c., from which the value of x in each case may be found by the methods already given, or more expeditiously by those given in Chapter VII. We have in the first equation x=0, for 10°=1, (Art. 26,) and in the others is 10*=11, 10*=12, &c. x=1.041 or 101.041-11 &c. 1 by Art. 56, we shall find a 1 10x: =-30 &c., and determining x from the x=-0.301 or 10-0.301 477 101000=3 101000-4 &c. 2 10: = 1 3 or &c. 1 100.301 1 100.477 x=-0.477 10-0.477- We see then that any number whatever may be considered as some power* of the number 10. For all numbers greater than unity the exponent of 10 will be positive, and for all numbers less than unity the exponent of 10 will be negative. 1 2 1 3 58. In the same manner any other number (except unity) may be made to express all numbers whatever by affixing to it a proper * The word power is frequently applied, as in the text, to all quantities 1 affected with exponents. In this sense, a" is a power of a whose degree m 1 n m ገፃ a power whose degree is This generalization of the word power becomes necessary by the extension of the notation of integral powers to negative and fractional exponents. n |