The Secret Trick to Solving Exponential Equations Fast!

Added: 2026-05-09

[00:00:00]I'll be showing you how to solve this Olympiad mathematics question.

[00:00:05]Do you believe that most students do not accept the solution to this equation?

[00:00:11]By the end of this video, you will know it.

[00:00:15]We have 3 to the power of x plus 3 to the power of x is equal to 33 to the power of x.

[00:00:21]So, we are asked to find the value of x.

[00:00:23]Follow me step by step as I provide a solution to this equation.

[00:00:28]Okay.

[00:00:29]Now, we shall have solution.

[00:00:35]Now, the question says 3 to the power of x plus 3 to the power of x to be equal to 33 to the power of x.

[00:00:46]Now, this is what we are going to do.

[00:00:49]Let's move 33 to the power of x to the left-hand side.

[00:00:54]Plus become minus when crosses equal sign.

[00:00:57]Then, we shall have 3 to the power of x plus 3 to the power of x minus 33 to the power of x is equal to 0.

[00:01:08]Now, look at this.

[00:01:10]Do you believe that 3 to the power of x is a factor of this, is also a factor of this, and is also a factor of this?

[00:01:19]Let me prove that this is true.

[00:01:21]The common factor here is 3 to the power of x multiplied by If this divide this, we shall have one.

[00:01:31]You put a plus sign.

[00:01:33]This divide this, we shall have one.

[00:01:35]Now, you put a minus sign.

[00:01:37]Let me show you that this can divide this without a remainder.

[00:01:41]If you have 33 to the power of x divided by 3 to the power of x.

[00:01:47]There's a law of indices that we are going to apply to solve this.

[00:01:51]And the law of indices state that if you have a to the power of b divided by c to the power of b, that this is the same thing as saying a over c all to the power of b.

[00:02:06]So, to apply this law to simplify this expression, we shall have 33 over 3 all to the power of x. Of course, 33 divided by 3 is 11. So, the whole of this become 11 to the power of x.

[00:02:22]What I'm trying to show you is that I'm going to prove that I'm proving that this can divide this without a remainder. And also, 3 to the power of x is a factor of 33 to the power of x.

[00:02:32]So, if this divide [clears throat] this, we are going to have 11 to the power of x. All this is equal to 0.

[00:02:40]Are you with me? Are you okay with this?

[00:02:43]Now, look at this. 1 + 1 is 2. So, so we shall have 3 to the power of x multiplied by 2 minus 11 to the power of x to be equal to 0.

[00:02:56]Okay. Now, we shall have 3 to the power of x to be equal to 0. That is the first equation.

[00:03:03]Now, in this place, x is a element of a non-empty set.

[00:03:08][snorts] So, we don't want this solution.

[00:03:12]Okay. So, we shall continue with 2 minus 11 to the power of x to be equal to 0.

[00:03:19]>> [snorts] >> To solve this, we shall have um 11 to the power of x to [snorts] be equal to 2. Because when you move minus 11 x to the right-hand side, this is what we shall have.

[00:03:32]Now, you cannot solve this like this because you cannot express the base to be the same.

[00:03:37]What we are going to do is to introduce um log to both side in base of 11.

[00:03:42]So, we shall have the log of 11 to the power of x base of 11 is equal to the log of 2 base of 11.

[00:03:53]This x multiply the whole of this log.

[00:03:57]We shall have the log of 11 or x log of 11 base 11 is equal to the log of 2 base 11.

[00:04:09]Log of the same base is always equal to 1.

[00:04:12]So, log 11 base 11 is 1.

[00:04:14]>> [snorts] >> x * 1 is x.

[00:04:17]So, we shall have x to be equal [snorts] to the log of 2 base of 11.

[00:04:24]And this happens to be the final answer.

[00:04:27]If this is the first time you come across this [snorts] video, air pulse to share, follow, and subscribe for more.

[00:04:34]Thank you. Oh, the power of