FROM ELEMENTARY AHiEBKA TO THE CALCULUS. 4I3 



assume that it certainly tended to a '.imit uni ormly, il at all, we 

 could make :v approach by being continually halved; then, since 



—^ ; = I («■' + i) , which is greater (or less) than i when a>i, 



^x X 



a" — 1 

 according as x is ±. /. [iov a > i) — —^ by this method of 



approach, continually diminishes when .v is + , and continually 



increases when % is - . Now -^ =«", which becomes 



X - X 



I when X =0, [and \t a < i, the function increases when x \> + 



and diminishes when .v is-1. 



Thus -^-— is defined as the value to which "' - converge 

 ±x 



(for we have proved that thev do converge) as x tends to 0. 



As there is nothing to indicate anv connection of this value 

 with previously known functions, and as it presumably depends on 

 a only, we give it a new name — the " (natural) logarithm " of a. 

 [The old word ' logarithm,' to base lo, as used in previous 

 arithmetic, is better not used — replaced by " ten index " : it 

 is ignored in this paper.] We proceed to prove from this 

 definition the fundamental properties of this logarithmic function, 

 and shall afterwards give a proof of its existence which does not 

 assume uniform convergence. 



Lt. {a'y~i I t a"' -1 



Since log «" = ^ n — ^ = ^ n :^ "" 



^ x^O X (ia; = U ax 



:. log rt" = a log a. (i). 



And log {ab) --= (assuming h -^a' ), log a'+' = (i + O- ^'^S « 



= log a + log a' = log a + log h . . . (ii). 



(i) includes log i/a= -log a and log i=0. 



The result obtained in the course of the proof : log ^ 



lies between N (a^ - i) and X (i - (/" >•). is often preferable to 

 expansion . 



Excursus. — To remove the unsatisf actor}' assumption of 

 uniformity of convergence we need to establish an inequality which 

 is quite elementary, but is unfortunately deferred in our text- 

 books while infinite series are (not very satisfactorily) discussed. 

 Viz.{'L + x)" lies between {i + nx) and [1 + nx. (i + a;)""'] for ali 

 values + and - of w and x, prov.ded always that we remember 

 that the theory of general indices assumes + bases to the indices. 



This proposition may be approached in two ways — (i) by 

 establishing the (infinite) Binomial Series for negative integral 

 indices, which is not difficult, and is perhaps desirable in an 

 elementary text-book ; or (ii) by proving that the geometric 

 mean of a number of quantities is less than the arithmetic mean. 

 Following this latter method, 



(i) if a +h' = a + b, a'h' = a' [a + b - a') = ab+ {a- a') {a - b). 

 Therefore, if a lies between a and b (and therefore b' also), the 

 product a'b' is greater than the product ab. 



