Which is Greater?? 🤯

Added: 2026-05-04

[00:00:00]This is a very interesting problem. We are asked to compare 3 raised to power pi and pi raised to power 3 and find out which one is greater. Before looking at the solution, can you solve it on your own and let me know your answer in the comments? Do not use calculators. Okay, since both numbers are not easy to compare directly, we have to use a smart method to simplify and compare them.

[00:00:28]You know what? Whenever we have expressions like a power b and b power a, we usually cannot compare them directly. So, a very useful method is to take natural logarithm or log with base e on both sides.

[00:00:44]That helps us bring the powers down and makes comparison easier. Thus, take log on both sides. Log of 3 power pi becomes pi multiplied by log 3. And log of pi power 3 becomes 3 multiplied by log pi.

[00:01:00]Now, do one thing. Divide both sides by 3 multiplied by pi.

[00:01:06]Since this is a positive number, the final comparison does not change.

[00:01:11]After dividing, we get log 3 divided by 3 on one side and log pi divided by pi on the other side.

[00:01:20]Now, here comes the magic. We will define a function to understand this better. Let f of x be equal to log x divided by x. So, by doing this, our problem now reduces to comparing f of 3, which will be this, and f of pi, which will be this.

[00:01:38]In the next step, we will study how this function behaves. To do that, we will use calculus and find the derivative of this function.

[00:01:47]Using the quotient rule for derivatives, the derivative of log x divided by x comes out to be 1 minus log x divided by x squared. Set the derivative equal to zero.

[00:02:00]1 minus log x equals zero. This gives log x equals 1. So, we get x equals e.

[00:02:10]So, the function has a critical point, which means either a maxima or a minima at e.

[00:02:16]To find whether it's a minima or a maxima, we will double differentiate this function. Again, using the quotient rule, we get the double derivative of f as this. Put x equals e here, and this becomes 2 minus 3 over e cubed, which is less than zero. Since the double derivative of f is negative, thus we get a maxima at x equals e. So, overall, this means that the function increases till x equals e and then decreases after e.

[00:02:51]Now, observe our numbers. This is 3, and pi is approximately 3.14.

[00:02:57]Both are greater than e, which is approximately 2.72.

[00:03:03]So, both values lie in the decreasing region of the function. Now, if a function is decreasing, then a larger input gives a smaller output. Since pi is greater than 3, so f of pi is less than f of 3. This means log 3 divided by 3 is greater than log pi divided by pi.

[00:03:24]This means pi multiplied by log 3 is greater than 3 multiplied by log pi.

[00:03:29]Hence, going back to the original comparison, we get 3 raised to power pi is greater than pi raised to power 3.

[00:03:38]And that's it. Isn't calculus super cool? If you enjoyed this video, please don't forget to like, share, and subscribe to our channel.

[00:03:47]So good.