Is π^e greater than e^π?

4 min readOct 24, 2023

Two numbers appear everywhere in the real world, π and e. π is also called Archimedes Constant. It appears mostly in circles and angles and has numerous implications for everyday objects and phenomena such as electricity(alternating current), Mobile Phone Technology (Wave Theory). e on the other hand is also seen everywhere. In chemistry, most fundamental reactions, including radioactivity, are first-order reactions in which the reactants are consumed exponentially. In certain circuits, similar phenomena can be observed while for example, when charging a battery or a capacitor. But what happens when you combine them? We have already seen one example of such a formula in the form of Euler’s Identity. Let us now find out which is greater, e^π or π^e. We will be doing so with two different methods, one using basic differentiation and the other using graphing and standard limits.

f(x)=x^{1/x}

This is a very famous function and is the function we shall use to find our answer. First of all, let us talk about the domain of the function. A function’s domain is the set of all places where the function is defined. Here, since f(x) is an exponential function, theory dictates that:

But since x is also in the fractional power, x cannot be 0. So the theory now dictates that:

So our domain is all real values of x, where x>0, and x<∞.

Now that we have found the function’s domain, we shall use it to our advantage.

Using Differentiation

Now let us take f(x) and apply it to e and π. We shall use a simple theorem in calculus. We shall not prove this theorem. The theorem states that:

Replace f with x and g with 1/x to get:

Now to find if f(x) is larger at x=e than at x=π, or vice versa, we shall use some differentiation to find out its monotonicity.

Applying the chain rule¹, we get:

Now use the quotient rule on lnx/x²³⁴

This now becomes:

Now apply the standard wavy-curve method⁵ to draw up a graph for f’(x)

To apply the wavy-curve, first, draw the number line. Then mark the critical points. Critical points are points where the value of the function becomes 0.

Clearly, the function becomes 0 when⁶:

This implies that:

Which means:

So the critical point for the function is x=e.

On the number line mark x=e and then check of numbers greater than e give a positive or negative value of f’(x).

e^{lnx/x} will always be a positive quantity. So just checking for the fraction will determine the sign of the term. Put x=e². We find that f’(x)<0 for all numbers greater than e. So that means f’(x)>0 for all numbers smaller than e. Thus e is the global maxima for the function⁷.

Thus we can imply that: f(x) increases in (-∞,e) and decreases in (e,∞).

Now,