LOGARITHMS. 



BEFORE reading the following pages become familiar with the 

 "Notation by Powers of io," page xv. 



The common or Briggs logarithm of a given number is the expo- 

 nent of that power to which 10 must be raised to produce the 

 number. Thus, 3 is the common logarithm of 1000, since io 3 = 1000. 



To multiply numbers together, add their logarithms. The sum is 

 tin- logarithm of the desired product. 



Proof. The product 10 x io 6 x X io m is i a+B - +m , since 

 powers of the same number may be multiplied together by adding 

 their exponents. Therefore, if A=io a , B=io b , -, M= io m , 

 A x B x X M = io a+b +-+ m . That is, a + b -\ + m is the log- 

 arithm of A x B x - X M. But a, 6, -, m are respectively log A, 

 log B, , log M. Hence, log (A X B x X M) == log A -f- log B 

 + +logJf. 



To divide one number by another, subtract the logarithm of the 

 latter from that of the former. The difference is the logarithm of 

 the quotient. 



Proof. Following the foregoing notation, A/B = io a /io 6 = 

 io a x io 6 = io a ~ 5 . Hence log A/B = a b = log A log B. 



Logarithm of a number of several figures. As i = 10 and io = io 1 , 

 the logarithm of i is o and of io is i, and the logarithm of any 

 number greater than o and less than io, that is, of any number in 

 the units' place (whether or not followed by a decimal fraction) is 

 less than j, that is, it is a fraction. It is expressed as a decimal 

 fraction. 



From the definition of a logarithm it is obvious that the logarithm 

 of any stated power of io is the index of the power; i.e. log io n 

 = n when n is any number, whole or fractional, positive or neg- 



xxiv 



