Harvard University Entrance Exam Question | Can you solve ?

Added: 2026-05-09

[00:00:00]Hello, welcome back once again. Today we're going to solve this interesting math problem. X to the^ of X is = 18.

[00:00:10]Now let us find out if X is an integer.

[00:00:14]Now let us start with X = 1. So 1^ 1 is equal to 1. So this is less than the right hand side. If x = 2 we get 2 rais^ 2 which is = 4 still less than the right hand side. If x = 3 we get 3^ 3 which is equal to 27. This is greater than the right hand side. So we can see that our equality sign changed from here from less than to greater than which is telling us something that our solution is somewhere between two and three.

[00:00:53]Right? Therefore, x is not an integer.

[00:00:56]Then we can say that 2 is less than x is less than 3. Okay. Now from the original problem we have x the^ x which is equal to 18. Let us ln both sides of this equation. We get the ln of x to the^ of x is equal to the ln of 18.

[00:01:21]Use this property of logarithm. The ln of a to the^ b is equal to b * the ln of a. The left hand side becomes x ln of x is equal to ln of 18. Use the property that a can be written in the form e^ of ln of a. So therefore we're going to write this x in that form. So here we have the ln of x time e^ of the ln of x.

[00:01:49]So this is equal to the ln of 18.

[00:01:52]Now let us recall the lambda function which is the product logarithm. The w of k * e ra to the^ of k is equal to k.

[00:02:04]So we can see that this left hand side has this pattern right. So if you take the w on both sides that is the w of ln of x * e^ of ln of x. This is equal to the w of ln of 18. So according to this property this simplifies to ln of x which is equal to the w of ln of 18. Now let us exponentiate both side using the base e. So we know e to the ln of anything simplifies to that. Therefore these both get cancelled leaving us with x which is equal to e rais^ of the w of ln of 18 and this is approximately equal to 2.803 8 03 663 2 4 6 and we can see that this is between 2 and 3 and this right over here is our correct answer. Thank you for watching.

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