THE TEN PILLARS 21 



(4) (a m ) n = a mn 



(5) (ab) m = a m b m 



(6) {a/b) m = a m /b m 



Fractional indices are called roots. Thus, a i = y/a, the square root of a; 

 and in general a'/"' = m \/~a, the m lh root of a. 



(7) a" = 1 



(8) a~" = 1/a" 



(9) a" = °o 



(10) a~' = 1/a" = 



Logarithms 



Let .4 = a". The index x, which tells how many times the base number a 

 must be multiplied by itself to give A, is defined as the logarithm of A to the base a. 

 In shorthand this statement is given by x = lbg a A, where "to the base a" 

 appears as a subscript to the abbreviated "logarithm." 



Logarithms are indices and must obey the ten Laws of Indices, just as any 

 other. For example: 



log AB = log A + log B 



log A/B = log A - log£ 



log A m = m log ,4 



A change of base from base a to base b turns out to be analogous simply 

 to a change of variable. In other words the logarithm to the base, a, is re- 

 lated to the logarithm to the base, b, by a constant, \o% b a. One is a linear 

 function of the other. 



This can be shown as follows. Suppose A = a" and A = b y , so that 

 a x = b y . Then log a A = log a b>, or x = y log a b. 



There are two systems of logarithms in daily use in biophysics, as in all 

 other science and technology: 



(a) Common logarithms, to the base \0(y = \0" for example), used to 

 simplify the manipulations of multiplication and division, based on rules (2) 

 and (3). The abbreviation is log, or log l0 . 



(b) Natural logarithms, to the base e (y = e x for example), where 

 e = 2.71828. . . . The base, e, and the functional relationship,}' = e", occur 

 over and over again in man's description of nature, and therefore will be 

 illustrated further. The abbreviation is In, or log f . 



