200 Algebra. 



ithms are entirely dependent upon the properties of exponents in 

 general, which have already been established. 



Among the fundamental properties of logarithms are these : 



The logarithm in any system of the base itself is i . 



For a^=a, 



that is, loga«=i. 



The logarith7n of u?iity i7i all systems is o. 



For ^"=1, 



that is, logai=o. 



Negative numbers have ?io logarithms. 



For in the equation a'-'—y, a is positive by supposition and l^y 

 the value of «' we mean that one of its values which is positive. 

 Hence J' cannot be negative. See Art. i. 



If we understand the same system of logarithms to be used 

 throughout, then the following four theorems hold. 



7. Theorem, The logarithm of the product of several niDubcr^ 

 equals the sunt of the loga? ithms of the separate factors. 



Let n and r be any two positive numbers and let 



log.j;>2=.r and log;,?"=.2'. (\} 



Then, by the definition of a logarithm, 



?^=<2' and r=a~. 

 Multiplying these equations together, member by member, 



nr==a-^"^~. 

 That is, \oganr=x-^3, 



or, from (i), log.., nr= log,,, n + log,, r. (a ) 



In the same way, if log;,^=z/, then 



7irs^=a^''*~^" . 

 That is, \ognnrs=\ogan + \ogar+\ogaS. 



8. Theorem. The logarithm of the quotient of tu'o numbers 

 equals the logarithm of the dividend minus the logaritlim of tlir 

 divisor. 



Let n and r be any two positive numbers and let 



\oZnn=x and log,,,r=5'. (i) 



Then, by definition, ^z=r<2'and r=a\ 



