Perhaps the most interesting feature of the function

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Perhaps the most interesting feature of the function log x is its be- haviour as x tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. The logarithm of x tends to infinity with x , but more slowly than any positive power of x , integral or fractional. In other words log x → ∞ but log x x α 0 for all positive values of α . This fact is sometimes expressed loosely by say- ing that the ‘order of infinity of log x is infinitely small’; but the reader will
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[IX : 202] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 449 hardly require at this stage to be warned against such modes of expression. 200. Proof that (log x ) /x α 0 as x → ∞ . Let β be any positive number. Then 1 /t < 1 /t 1 - β when t > 1, and so log x = Z x 1 dt t < Z x 1 dt t 1 - β , or log x < ( x β - 1) /β < x β /β, when x > 1. Now if α is any positive number we can choose a smaller positive value of β . And then 0 < (log x ) /x α < x β - α ( x > 1) . But, since α > β , x β - α 0 as x → ∞ , and therefore (log x ) /x α 0 . 201. The behaviour of log x as x +0 . Since (log x ) /x α = - y α log y if x = 1 /y , it follows from the theorem proved above that lim y +0 y α log y = - lim x + (log x ) /x α = 0 . Thus log x tends to -∞ and log(1 /x ) = - log x to as x tends to zero by positive values, but log(1 /x ) tends to more slowly than any positive power of 1 /x , integral or fractional.
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[IX : 202] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 450 202. Scales of infinity. The logarithmic scale. Let us consider once more the series of functions x, x, 3 x, . . . , n x, . . . , which possesses the property that, if f ( x ) and φ ( x ) are any two of the functions contained in it, then f ( x ) and φ ( x ) both tend to as x → ∞ , while f ( x ) ( x ) tends to 0 or to according as f ( x ) occurs to the right or the left of φ ( x ) in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with log x , which tends to infinity more slowly than any of the old terms. Then log x tends to more slowly than log x , 3 log x than log x , and so on. Thus we obtain a series x, x, 3 x, . . . , n x, . . . log x, p log x, 3 p log x, . . . n p log x, . . . formed of two simply infinite series arranged one after the other. But this is not all. Consider the function log log x , the logarithm of log x . Since (log x ) /x α 0, for all positive values of α , it follows on putting x = log y that (log log y ) / (log y ) α = (log x ) /x α 0 . Thus log log y tends to with y , but more slowly than any power of log y . Hence we may continue our series in the form x, x, 3 x, . . . log x, p log x, 3 p log x, . . . log log x, p log log x, . . . n p log log x, . . . ; and it will by now be obvious that by introducing the functions log log log x , log log log log x , . . . we can prolong the series to any extent we like. By putting x = 1 /y we obtain a similar scale of infinity for functions of y which tend to as y tends to 0 by positive values.
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