December 12th, 2017, 09:34 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,234 Thanks: 2412 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, 03:14 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 18,704 Thanks: 1530 
How would you solve that "from first principles"?

December 12th, 2017, 06:12 PM  #13 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,234 Thanks: 2412 Math Focus: Mainly analysis and algebra 
The link contains a derivation.
