L O G 



both of which last series converge very fast. 

 Ex. If N = 2, Lll - I ; .'. log. 2 - .3010300. 



In hyp. logarithms m = 1, in the common system z 9 - - ^ ~- 



,43424948. And since different systems of logs, are as their moduli, if any 

 common log. be divided by this modulus, it gives the corresponding hyp. 

 log. ; or if any hyp. log. be multiplied by it, it gives the corresponding" 

 common logarithm. 



3. Given a logarithm, to find its number. 

 Let 1 + x =. No., y its log. m the modulus. 



then 1 -f x = 1 + -S- + J. + ^L. + & c . 



r m T 2 m T 2. 3. m* T 



If m = 1 ( l+o;-l+3/ + ^- + ^T 4- &c - = No - whose hyp. log. 



4. Modular ratio is the ratio of which the modulus is the measure, or 

 the number of which the modulus is the logarithm, and = 1 + 1 + 



+ - + &c. : 1 ; or 2.7182818 : 1 ; which is therefore the same for 

 every system, being independent of m and y. 



Hence in Napier's or hyp. logs., where the modulus is 1, the log. of 

 2.7182818 is 1 i in Brigg's or the common system, log. 2.7182818 ia 

 ,43424948. 



Hence also since in every system the log. of the base is 1 ; 2.7182818 is 

 the base of Napier's logs. ; in Brigg's the base is 10. 



In general if a = base of any system, whose modulus is m, m = h ^ ^. 



The following Table of Logarithmic series will be found useful on various 

 occasions. 



1. Log. a = ~ x (-l) -i (-!)* + 1 (a- 1) 3 - &c. 



