Olympiad Mathematics | Find X | Can You Solve This?

Added: 2026-05-23

[00:00:00]Hello everyone, you are welcome. Today we have a very interesting exponential math problem.

[00:00:07]What is the value of x here in x raised to power x is equal to x squared?

[00:00:14]So let's start our solution. So how can we solve this beautiful exponential and algebra math problem?

[00:00:19]First of all, here we will divide both sides by this one expression. So this one equation will become this is simply x raised to power x divided by x squared is equal to and the right hand side will become x squared divided by x squared.

[00:00:39]And in the right hand side, this x squared and this x squared will be cancelled. And the left hand side in both numerator and denominator, both the bases are same.

[00:00:49]Here we will use same base exponential method property. That is here we can write a raised to power m divided by a raised to power n as a raised to power m minus n.

[00:01:03]So using this exponential method property here, this left hand side will become this will become x raised to power x minus 2 is equal to and this will become 1.

[00:01:17]Here we will take natural log on both sides. So this expression and this one equation will become ln of x raised to power x minus 2 is equal to ln of 1.

[00:01:33]And in the left hand side, we will use a log property. That is here we can write natural log of x raised to power a as a times natural log of x.

[00:01:47]So using this log property here, this left hand side will become this will become x minus 2 times natural log of x is equal to here natural log of 1.

[00:02:02]This is equal to 0.

[00:02:04]So, we will replace this with 0.

[00:02:07]Here the product of these two expression is 0. So, here either this expression will be 0 or this one will be 0.

[00:02:14]So, from here we will get x - 2 is equal to 0.

[00:02:20]Or ln of x is equal to 0.

[00:02:26]So, let's solve this one equation first.

[00:02:27]Here we will take this two to the right hand side. This gives x is equal to 2.

[00:02:33]So, this is our first real solution.

[00:02:36]And we will try to find out the value of x from this one equation. So, how can we solve this one equation?

[00:02:41]Here we will take e in the base in both sides.

[00:02:45]So, this will become this will become e raised to power ln of x is equal to e raised to power 0.

[00:02:54]And here we know that e raised to power ln of a is equal to a and e raised to power 0 is equal to 1.

[00:03:07]So, using this result here this one equation will become this will become x is equal to 1.

[00:03:15]So, this is the second real value of x.

[00:03:17]So, finally here we have two values of x in this problem.

[00:03:21]x is equal to 1 and x is equal to 2.

[00:03:27]And we will try to verify these values of x that as these value of x verify this interesting algebra and exponential equation or not.

[00:03:37]So, first we will try to verify x is equal to 1.

[00:03:42]To verify x is equal to 1 here we will write our equation again.

[00:03:47]Our equation is simply x raised to power x. This will become 1 raised to power 1 is equal to 1 raised to power 2.

[00:03:58]1 raised to power 1 is simply 1 is equal to and 1 squared is again 1.

[00:04:03]Here both sides are equal, so it means that x is equal to 1 is the exact and correct value of x.

[00:04:10]And we will try to verify x is equal to 2.

[00:04:15]So, our equation will become that will become 2 raised to power 2 is equal to 2 raised to power 2.

[00:04:23]2 raised to power 2 or 2 squared is simply 4 is equal to 4.

[00:04:29]Again, both sides are equal, so it means that x is equal to 2 is also the exact and correct value of x.

[00:04:37]So, finally, x is equal to 1 and 2 are the exact and correct solution of this interesting algebra and exponential math.