e^π or π^e. Which one is bigger?

No calculator allowed. They're surprisingly close: about 23.14 and 22.46, less than one apart. The trick: take logs, and the question becomes a comparison of one function, f(x) = ln x / x, at two inputs. That curve has a single peak, and it sits exactly at e. We'll see why the peak is at e (with a line swinging out from the origin), why 2⁴ = 4², and how the whole thing turns into the x-th root of x. 0:00 Pick a side 0:23 The bigger base can lose 0:45 Make the powers comparable 1:41 One curve answers the question 2:28 Why exactly e? 3:40 When neither wins 4:13 The winning root