EXPONENTIAL AND LOGARITHMIC SERIES 15 



But (1 4- z)" can be expanded by means of the Binomial 

 Theorem. 



Equating coefficients of y in (1) and (2) 



log e (l + z)=s-^+|--^.. . (8) 



This series can be conveniently used for values of z less than 1, 

 and it can therefore be employed as a means of calculating the 

 logarithms of numbers between 1 and 2. 



Replacing z by z we have 



~2 ~3 ~4 



1^-4., -.-._.. (4) 



a series which can be used for the calculation of the logarithms 

 of numbers and 1. 



Subtracting (4) from (3) gives 



log.(l + z) - log e (l - z) or log, ji5 = 2Jz + i- + y + . ' . .} 



1 + z n + I 1 



putting - - = - or z = - r the series becomes 

 b 1 - 2 n 2n+l 



n+ 1 f _ 1_ 



ge n \2n+l 



+ 



(2n+ l) 3 5(2n+ 



a series in which the terms rapidly become smaller and smaller 

 as n is made larger, and it can be used for the calculation of the 

 logarithms of numbers greater than 2. 



For log(n + 1) - log e n - 2 



. 



when n = 1 log,2 = 2 {J + ^3+ ^ + } 



= 0-6931 

 when n = 2 Iog e 3 - Iog e 2 = 2^ + ^ + ^ . . .} 



= 0-4055 



Then Iog e 3 = 1-0986 



Now Iog 4 4 = 2 Iog e 2 = 1-3862 



when n = 4 Iog e 5 - Iog e 4 = 2J1 + ^ + JL . . .} 



= 0-2231 

 Then log,5 - 1-6093 



