e^pi vs pi^e

Added: 2026-06-04

[00:00:01]e^pi vs pi^e, which one's bigger? Well, check this  out. I will be using a solution from a viewer from eight years ago. E to the x is greater than 1 + x.  And although this is true for all x, I will just write down 4x is greater than zero because the  proof for this is easier because if you look at e to the x, the power series suspension for that,  we get this thing here. And if you just consider the past dx, if you get rid of all these terms,  then you see e to the x has to be greater than 1 plus x. Then we are going to let x equal<unk> / e  minus one. And notice this is indeed positive. So we can plug that in. E to the<unk> / e minus one  has to be greater than 1 +<unk> / e minus one.

[00:00:55]This and that cancel. And then let's multiply both  sides by e. This times that add the exponents. We get e to the pi over e is greater than just  pi. Then we just have to raise both sides to the e power. And when we multiply exponents,  we get e to the pi. And then when we have this, it's just pi to the e. And right here, we still  maintain the same inequality because the insides are positive. When we raise that to the positive  power, it's an increasing function. That's it.