Gentle when x is very large and very steep when x is

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gentle when x is very large, and very steep when x is very small. Examples LXXXII. 1. Prove from the definition that if u > 0 then u/ (1 + u ) < log(1 + u ) < u. [For log(1+ u ) = Z u 0 dt 1 + t , and the subject of integration lies between 1 and 1 / (1 + u ).]
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[IX : 198] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 447 2. Prove that log(1 + u ) lies between u - u 2 2 and u - u 2 2(1 + u ) when u is positive. [Use the fact that log(1 + u ) = u - Z u 0 t dt 1 + t .] 3. If 0 < u < 1 then u < - log(1 - u ) < u/ (1 - u ). 4. Prove that lim x 1 log x x - 1 = lim t 0 log(1 + t ) t = 1 . [Use Ex. 1.] 198. The functional equation satisfied by log x . The function log x satisfies the functional equation f ( xy ) = f ( x ) + f ( y ) . (1) For, making the substitution t = yu , we see that log xy = Z xy 1 dt t = Z x 1 /y du u = Z x 1 du u - Z 1 /y 1 du u = log x - log(1 /y ) = log x + log y, which proves the theorem. Examples LXXXIII. 1. It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from log x . For when we differentiate the functional equation, first with respect to x and then with respect to y , we obtain the two equations yf 0 ( xy ) = f 0 ( x ) , xf 0 ( xy ) = f 0 ( y ); and so, eliminating f 0 ( xy ), xf 0 ( x ) = yf 0 ( y ). But if this is true for every pair of values of x and y , then we must have xf 0 ( x ) = C , or f 0 ( x ) = C/x , where C is a constant. Hence f ( x ) = Z C x dx + C 0 = C log x + C 0 , and it is easy to see that C 0 = 0. Thus there is no solution fundamentally distinct from log x , except the trivial solution f ( x ) = 0, obtained by taking C = 0.
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[IX : 199] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 448 2. Show in the same way that there is no solution of the equation f ( x ) + f ( y ) = f x + y 1 - xy which possesses a differential coefficient and is fundamentally distinct from arc tan x . 199. The manner in which log x tends to infinity with x . It will be remembered that in Ex. xxxvi . 6 we defined certain different ways in which a function of x may tend to infinity with x , distinguishing between functions which, when x is large, are of the first, second, third, . . . orders of greatness. A function f ( x ) was said to be of the k th order of greatness when f ( x ) /x k tends to a limit different from zero as x tends to infinity. It is easy to define a whole series of functions which tend to infinity with x , but whose order of greatness is smaller than the first. Thus x , 3 x , 4 x , . . . are such functions. We may say generally that x α , where α is any positive rational number, is of the α th order of greatness when x is large. We may suppose α as small as we please, e.g. less than . 000 000 1. And it might be thought that by giving α all possible values we should exhaust the possible ‘orders of infinity’ of f ( x ). At any rate it might be supposed that if f ( x ) tends to infinity with x , however slowly, we could always find a value of α so small that x α would tend to infinity more slowly still; and, conversely, that if f ( x ) tends to infinity with x , however rapidly, we could always find a value of α so great that x α would tend to infinity more rapidly still.
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