Harvard University Entrance Exam Question | Can you solve ?

Added: 2026-05-31

[00:00:01]Hello, welcome back once again. Today we're going to solve the following given interesting equation.

[00:00:07]X raised to the power of X is equal to 16. In this video, we're going to find the value of X that will satisfy this equation. So let's get started.

[00:00:18]Now, let us find out if X is an integer.

[00:00:22]So if X is equal to 1, we get 1 raised to the power of 1. This is equal to 1, which is less than the right hand side.

[00:00:31]If X is equal to 2, we get 2 raised to the power of 2, which is equal to 4.

[00:00:37]This is less than the right hand side.

[00:00:40]If X is equal to 3, we get 3 raised to the power of 3, which is equal to 27.

[00:00:46]This is greater than the right hand side.

[00:00:50]Now we can see from here our equality sign change, right? From less than to greater than. So it tells us that our solution lies between 2 and 3.

[00:00:59]Hence, we say 2 is less than X and X is less than 3. Awesome.

[00:01:10]Now here we have X to the power of X is equal to 16.

[00:01:17]Let us ln both sides. We get the ln of X to the power of X is equal to the ln of 16. Use the power rule of logarithm. This becomes X times ln of X, which is equal to ln of 16.

[00:01:35]Now use this following property. A can be written in the form E raised to the power of ln of A.

[00:01:43]So therefore here we have ln of X then multiplied by E raised to the power of ln of X. So this is equal to ln of 16.

[00:01:54]Now we can use the Lambert W function on both sides of this equation.

[00:01:59]So, this is the Lambert W function.

[00:02:03]So, it has a very nice property. The W of K times E raised to the power of K is equal to K.

[00:02:12]With this property, the left-hand side simplifies nicely to ln of X, which is equal to the W of ln of 16.

[00:02:25]Now, from here we're going to exponentiate both sides using the base E.

[00:02:30]So, here this will be E to the power of ln of X and here E raised to the power of the W of ln of 16. So, this both get cancelled. So, here we have X is equal to E raised to the power of the W of ln of 16.

[00:02:46]So, here this will give us our correct value for X.

[00:02:50]That is X is approximately equal to 2.

[00:02:54]745 368 023 57.

[00:03:05]So, this right here is our correct answer.

[00:03:08]Thank you for watching. If you enjoyed the video, please kindly subscribe to this channel. Also, like, comment, and share.