/ 

 Logarithms. 197 



2. Theorem. 77ie expression a"" can be made to differ from i 



by less fhaji a^iy assigned quantity if x be sufficiently increased. 

 Suppose it be required to increase x so that 



where a' stands for an assigned positive number. Then we must have 



(i^dr->a, (2) 



or, by the binomial theorem, 



I . 2 



It is easy to see that i -^xd can always be made greater than a, 

 however small d may be. Much more, then, will the left member 

 of ("3 j be greater than a. In fact, the inequality 



i-j-xd^a 

 will hold i f xd^ a—i, 



or if .r> ~-y- . 



a 

 1 

 Hence to make a ■'' less than i-\-d take 



1 

 Example: Find .v such that 10* <i.oooi. 



Here <^/=.oooi and ^=10; whence 



Q 



x^ , or goooo. 



.0001 



3. Theorem, The expression a"" is a cofitinnous function of x. 

 Suppose a^^=y and let x take on any increase, s, and suppose 



the corresponding value of _i' bej' + /, so that 



^a--'=y-^t. (i) 



We are to prove that as x passes continuously from x to x-\*s 

 that y passes continuously from y to y-\-t; that is, as x changes 

 from X to x-\-s by passing over every intermediate value that y 

 changes from r to r+/ by passing over every intermediate value. 

 The equation «*"=r-f ^ may be written 



^"«'=J+^, , (2) 



and .since «'=j', this may be written 



or a''(a'—i) = t. (^) 



