Exponential function no relation R
The real exponential function is a bijection from 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
   to 
  
    
      
        (
        0
        ;
        ∞
        )
      
    
    {\displaystyle (0;\infty )}
  . Its inverse function is the natural logarithm, denoted 
  
    
      
        ln
        ,
      
    
    {\displaystyle \ln ,}
   
  
    
      
        log
        ,
      
    
    {\displaystyle \log ,}
    or 
  
    
      
        
          log
          
            e
          
        
        ;
      
    
    {\displaystyle \log _{e};}
   because of this, some old texts refer to the exponential function as the antilogarithm.As functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
no relation