Part 5/5:
Thus, using calculus, logarithmic manipulation, and foundational exponential properties, we have effectively shown that ( e^\pi > \pi^e ). This exploration not only reinforces mathematical principles but also highlights the beauty and interconnectedness of mathematical constants. Such inequalities serve not merely to satisfy curiosity but also to deepen our understanding of the underlying structures that govern mathematics. Stay tuned for more explorations into the fascinating world of math!