THE TEN PILLARS 



23 



The series typified by e kx is the only functional relationship in all of mathe- 

 matics for which its instantaneous rate of change at a value of x is exactly 

 proportional to itself. That is, it is the only function for which both 



y a e kx (1-4) 



and 



dy/d.v « e kx (or « y) (1-5) 



are true. 



For completeness, if the proportionality constant in Eq. (1-4) is intro- 

 duced, 



y =y ^ ---(1-4') 



and 



dy/dx = ky e kx ,__(l-5') 



or 



dy/dx = ky 



This, however, explains the importance of e x in mathematics. The im- 

 portance in biophysics is that a great many naturally occurring phenomena 

 are observed to behave according to Eq. (1-5'): many chemical reactions, 

 growth, diffusion processes, radioactive decay, radiation absorption phe- 

 nomena, etc. (Figure 1-8). 



TIME 



Figure 1-8. Two Exponential Relationships: 

 Growth (positive k), and Decay (negative k). 



For example, let y be the number of atoms of a given sample which give 

 out a radioactive emanation (alpha, beta, or gamma ray), and x be the time. 

 Eq. (1-4') says that the rate of emanation is always proportional to the num- 

 ber of atoms which are left and are capable of disintegrating, a statement 



