Logarithms. 2ii 



But if we substitute 2X in place of ji: in equation (^ij we obtain 

 «-=I-|-2BJl:^-4C;t:"-{-8DJt-3^-I6E-^-'^- . . . (2,) 



Therefore, equating like powers of ^ in equations (2) and (^3^, 

 we obtain 



B' B^ B^ 



B=B; C= ; D= - ; E= ; etc. 

 2 J 14 



Whence, on substituting these values of B, C, D, etc., equation 



(\) becomes 



!_2 1 3 1 4 

 Now, there must exist some quantit}^ e, at present unknown 

 in value, such that 



c''=a. (5) 



or, m other words, such that 



loge<2=B. (6) 



Substituting loge^^- for B throughout equation (/^) we obtain 



\2 [3 1 4 



which is called the Expo7icntial Theorem or Series. 



29. T^o FIND THE Vai^ue OF THE BASE e. The base a in the 

 last article is any chosen positive quantity not i, and its value is 

 therefore at our disposal. Hence in the exponential series (equa- 

 tion 7) we may put a=e, so that loge^ becomes loge^ ; that is, i. 

 Equation (']) then becomes 



X^ X^ X* 



^'='+-+,y+i:3 + |4 + --- ^'^ 



This important result is convergent for all values of x, (see 

 XIV, Art. 12, Ex. 13,) and consequently the equation is true 

 when x= i . Therefore we have 



By taking a sufficient number of terms of this series we may 

 approximate the value of e to any desired degree of accuracy. 

 Thirteen terms of the series give ten places of decimals correctly 

 and we have 



^=2.7182818284 . . . C3; 



