Q:  Why does using magnitudes turn the exponential relationship
          into a linear one?


           Magnitudes are defined using the logarithm of intensity:

                  mag   =   const  -  2.5*log10 ( Intensity)




  Q:  What is the relationship between the coefficients f and k?

                                          I 
              (m - m0)  =  -2.5 log10 ( ----- )
                                          I0
   

        
           But in this case,


                     I        -fX
                   -----  =  e
                     I0


           Thus

                                          -fX                 1         -fX
                (m - m0)  =  -2.5 log10 (e    )  =  -2.5 * ------ * ln(e    )
                                                           ln(10)

                          =  -1.09 (-fX)


                          =   1.09 * fX

            And so

                     m    =  m0  + 1.09*fX   =   m0  +  kX


            Therefore,

                        1.09*f  =  k

            or, to a decent approximation

                             f  ~  k