"-H ,+ - + - -+ . . . ! = I.6094379- 



9 3-9' 5-9^ 7-9' J 



214 Algebra. 



To find logeS, put ^=4 in equation (5). We then have 



loge5= 1-3862944 + 2 



In like manner the logarithms of all numbers may be founcL 

 The logarithms of composite numbers need not be computed by 

 the series, since the logarithm of any composite number can be 

 found by adding the logarithms of its component factors. 



34. RKiyATioN Between the Logarithms of the Same 

 Number in Different Systems. 



Consider the systems whose bases are a and e. Then if n is any 

 number, we wish to find the relation between logon and log;, 71. 



Let x=^logen and j'=lognn. 



Then «=<?" and n=a''; 



whence ^' — a^. (i) 



Therefore a^e^' . (2) 



If we write this in logarithmic notation we have 



logea-=~, (2,) 



or, substituting the values of x and y, we obtain 



1 ^ogen 



log,..=i^g^;„. (4)- 



Therefore logun = , - loge^^ Ts/ 



logea 



which is the relation between log..,w and log en. 



35. Modulus of Common Logarithms. If in equation f^)' 

 above we understand e to reprCvSent the Naperian base and a the 

 common base, then equation (^) becomes 



^^^ ^=iog;io^^^^^^' ^'^ 



But logeio=loge2 + loge5 = (by Art. 33) 2.3025851 and ~ 



log (. I o 



= .43429448 Therefore representing .43429448 by M we have 



log 7i=M logon. (2) 



The decimal represented by M is known to 282 decimal places. 



and is called the Modulus of the system of common logarithms. 



