Which one is greater e raised to the power π ( e^π) or π raised to the power e ( π^e)?

(For understanding the number e, you may wish to click on this link: e for eternity. And to understand the number π, you may wish to click on the link: True life of π.)

Today's Question: Which one is greater: e raised to the power π (e^π ) or π raised to the power e (π^e)? 

This a classical question.The answer can be found immediately by using Calulator or by taking logarithm of both the numbers and then comparing. 

However, let us try to guess the answer without calculating and without using Logarithm.

                Approach (1) 

In e^π , the base is smaller (e= 2.718), but the exponent is larger (π = 3.14). And in π^e , the base is larger, but the exponent is smaller (e).

But note that both the numbers are in the vicinity/ proximity of 3.

So which effect wins: the larger base or the larger exponent?

Let us try to understand with the example of 2³ and 3². Both these bases/ exponents are 3 or less than 3. Here 3² is greater than 2³. In this case the number with greater base is greater.

Now  take example 3⁴ and 4³, or 4⁵ and 5⁴,  or 5⁶ and 6⁵ so on and so forth. In all these cases numbers with higher exponents are greater.

The key idea here is: For numbers  greater than 3, increasing the exponent has a  stronger effect than increasing the base by a similar amount.

Both changes increase the value, but the gain from increasing the exponent is  larger than the gain from increasing the base. 

From this approach , it is quite safe to conclude that e^π is the greater number as the exponent π is greater than 3. So e^π ends up ahead.

                Approach (2)

Imagine money growing by compound interest: The base is like the interest rate and the exponent is like the number of years you let it compound.

A slightly lower interest rate over more years can outperform a slightly higher interest rate over fewer years. Here, the "extra compounding time" provided by the exponent π more than compensates for the slightly smaller base e.

So, in plain language: Although π is bigger than e, the advantage of having the larger exponent (π instead of e) outweighs the advantage of having the larger base.

Therefore, e^π  is larger than π^e .

Concluding Part:   

A fascinating historical note is that π and e arise from completely different origins.

π comes from geometry: the ratio of a circle's circumference to its diameter. Whereas e comes from growth and continuous compounding, calculus, and logarithms. But they both gel so well in different situations.

We will discuss more about the gelling of e and π soon in a different context.