December 12th, 2017, 10:34 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,508 Thanks: 2513 Math Focus: Mainly analysis and algebra 
The first principles approach would be to solve the functional equation $$f(xy)=f(x)+f(y)$$ You have to arbitrarily pick one of the solutions, but the choice of $f'(1)$ is reasonably natural and gives you a definition of $e$. The Glorious Logarithm 
December 12th, 2017, 04:14 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 19,951 Thanks: 1842 
How would you solve that "from first principles"?

December 12th, 2017, 07:12 PM  #13 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,508 Thanks: 2513 Math Focus: Mainly analysis and algebra 
The link contains a derivation.
