SOLUTION: We know that the graph of a logarithmic function is symmetrical to the graph of its corresponding exponential function with respect to the line y=x. Will the two graphs intersect w

Looking at base = e, and ln (the natural log)...



A graph showing , , and  is a quick way to see they will never intersect.





Aside from that... for , clearly  as  while .  For the half-plane x>=0, one can write the Taylor series expansion for :  

   = 



Pull out the first two terms from the RHS:

   = 



So clearly  for  and combined with the first part,  for all x.





To prove   <    (here the domain of x is x>0, as ln(x) is not defined for  ),  we can use the result from above as the starting point:



       

Take ln() both sides:

        > 

        >    , x>0