198 Algebra. 



Now, by the last theorem, by taking .? small enough a' may be 

 made to differ from i by an amount as small as we please. Hence 

 in equation ( ^) i may be made as small as we please by taking <r 

 small enough. That is, the difference between two successive 

 values of a^ can be made as small as we please. Therefore it is a 

 continuous function of x. 



4. It follows directly from the above theorem that for every 

 positive value whieJi may be assi<^7ied to y in the equation a''=y, a 

 corresp07iding vatue of x exists ivhieh 7cill satisfy the equation. 



For the last article shows that as x is increased continuously 

 from the value o without limit that r increases continuously from 

 the value i without limit. That is, y may have every value 

 greater than i. It is also seen that as .r is decreased continuously 

 from the value o without limit that r decreases continuously from 

 the value i . That is, y may have every fractional value. 



The above shows that if any value be assigned to j' in the equa- 

 tion ^"=1' that a value of .v exists which will satisfy it, but it 

 does not explain how to find that value. Thus it does not show 

 how to find X in the equation 10' = 5. The method of finding this 

 will be explained later. 



5. DefinitioNvS. In the equation ^' = j', where a is some 

 chosen positive number not i ; 



The constant quantit}^ a is called the Base. 



The quantity^ is called the Fixponential of .r to the base a. 



The quantity x is called the Logarithm of j' to the base a, and 

 is written a-=logar. 



The use of the w^ord logarithm may be kept in mind by remem- 

 bering this sentence : In the equation <2' =-j', x is called the Ex- 

 ponent of the power of a or the Logarithm of y. 



Of course the two equations 



a'=y (i) 



.r=logaJ' (2) 



express the same truth respecting the relation between .v and y. 

 The second equation uses the logarithmic notation and is always 

 to be interpreted by means of the first equation. 



If in the equation «'=_>', w^here a is some positive number not 

 I, different values be assigned to y and the corresponding values 



