log{e}(pi) = 1/log{pi}(e)
So we need to show x + 1/x > 2 whenever x is not equal to 1. (Since pi and e are not equal, neither logarithm is 1.)
Take the function y = x + x^(1). At 1 it has the value 2.
Its derivative is 1  x^(2), and at 1 that is zero, so the function is at a minimum or maximum.
The second derivative is 2x^(3), which is 2 at x=1, which is positive, at any x, so x=1 is a minimum, and in fact the only relative extremum. Therefore any other value of x + 1/x is larger than 2 for any real x.

Posted by Charlie
on 20170227 10:10:00 