Logarithms. 203 



positive or negative, special names are given to each part. The 

 positive or negative integral part is called the Characteristic of the 

 logarithm. The positive decimal part is called the Mantissa. 



14. The following table is self-explanator>^: 



ID"* =10000, whence log 10000=4 

 lo"* =1000, " log 1000=3 



10- =100, " log 100=2 



10' =10, " log 10=1 



10" =1, " log 1=0 



io~'=.i, " log .1 = — I 



io~"=.oi, " log .01 = — 2 



io"~^=.ooT, " log .001 = — 3 



io~'*=.oooi, " log .0001 = — 4 

 Here we observe that as the numbers pass through the series 

 loooo, 1000, 100, 10, etc., the logarithms pass through the series 

 4, 3, 2, I, etc.; that is, continuous division of the number b}- 10 

 corresponds to a continuous subtraction of i from its logarithm. 

 This can easily be shown to hold in any case. 



15. Theorem. Multiplying any number by 10 increases the com- 

 mon logarithm by- i, and dividing any number by 10 decreases its 

 common logai^ithm by i. 



Let r be any number and x its common logarithm. Then 



logj'=.r, 

 or 10' = )'. 



We are to prove log ioi'=.r-f i, 



and log yVvj'^-^^i- 



By formula (a). Art. 7, 



log ioj'=log j'.+ log 10. 

 But log 10=1 (Art. 6) and log i'=x. 



Hence, substituting, log 10 ]'=.rH-i. 

 Also by formula (b), Art. 8, 



log TVj=log_r— log 10. 

 That is, log ^i^i'=.v— T. 



