lyOGARlTHMS. 20] 



Consequently -=--=«' 



r a~ 



Therefore, by definition, 



n 



or, by equation (i), 



log« - I =log„7z-logar. (b) 



9. Theorem. The logarithm of a powei' of a number is equal 

 to the logarithm of the number multiplied by the exponent of the 

 power. 



I^et 71 be any number, and let Xo'gan^x. Then, by definition, 



n^a"" . 

 Consequently n^=a^''. 



Therefore, by definition, log/j ;z^=/>j;. 

 That is, \ogan^—p \ogan. (e) . 



10. Theorem. The logaritJmi of any root of a number is equal 

 to the logarithm of the fuunber divided by the index of the root. 



Let 71 be any number, and let \o%an---x. Then, by definition, 



Consequently \/n=a'^ . 



Therefore, by definition, 



log„(V«)=^. 

 That is, log,(V«)=^°^«". (d) 



1 1 . Theorem . If several n um bers are in geometrieal progression , 

 their loga7dth77is are in a7'ith77ietical p7'ogressio7i. 



Let the numbers which are in geometrical progression be rep- 

 resented by 



71, nr, 7ir^y nr^, . . . 



Then their logarithms to the base a form the series 



log,, n , log,, n 4- log,, r, log,, n -\- 2 log,, r, log,, w + 3 log,, ;',... 

 which is an arithmetical progression with the common difference 

 equal to log,,r. 



