14 PRACTICAL MATHEMATICS 



great, but ( 1 + - ) can be expanded by means of the Binomial 



Theorem. 



/ ' l\* /1\ nas(nxI)/l\* 



1 1-1 ) =1 +nx{ - ' ' 

 V n/ \n 



n/ \y \n/ |3 \n 



i\ ( r 



X\X ) X\X II X 



\ . n/ \ fi/\ n> 



= 1+ a? + + p- + . . . 



!_. LI 



Making n infinitely great, we get 



By giving x different values we can use this series to calculate 

 the corresponding values of e K . Thus when x = - 



I 1 1 1 



z>* i i i i i 



4 

 6 

 8 



10 

 12 

 14 



2 2 | 2 2 3 |_3 



21-000000 

 500000 



-125000 

 020833 

 002604 

 000260 

 000022 

 000002 



1-648721 

 Then e 2 = 1-64872 correct to six significant figures. 



If e* = 1-64872, then log e 1-64872 = i and therefore we can 



2 



use the series for e x as a means of calculating numbers if we are 

 given their logarithms to the base e. We can thus consider " e " 

 to be the base of a system of logarithms, and such a logarithm is 

 called the Napierian or hyperbolic logarithm. 



To change the base of a system of logarithms we may put x = cy. 



Then e = e ey = a? 



where e c = a or c = log e a 



/t'Ji/2 /**>4/3 



2 ^~ *l 



and a" = 1 + y log e a + JL- (log e a) 2 + -^- (log e a) 3 + . . . 

 Replacing a by 1 + z 



(l + z)'-l + y log e (l + z) + -{log e (l +z) ) -f . . . (1) 



I* 



