Can You Find ALL Real Solutions to This Tricky Exponential Equation?

追加: 2026-05-26

[00:00:00]Let me show you how to solve this Olympiad mathematics question. Welcome to my channel.

[00:00:07]Now, the question says 2 to the power of x minus 1 is equal to 3 x plus 1. We are asked to find the value of x.

[00:00:14]In this video, you are going to learn the practical way of solving this.

[00:00:17]It is very simple. I want you guys to follow me step by step as I am solving this together.

[00:00:23]Now, do you believe that I can write this as 2 to the power of x minus 1 to be equal to 3 x plus 0 plus 1 because if I add this to 1, I'm still having one. Nothing change.

[00:00:38]I want to rewrite this 0 as minus 3 plus 3.

[00:00:43]So, this become 2 to the power of x minus 1 to be equal to 3 x plus No, this is minus 3 plus 3 because minus 3 plus 3 is 0. That is plus 1.

[00:00:59]Now, watch this.

[00:01:02]3 plus 1 is 4. This become 2 to the power of x minus 1 to be equal to 3 x minus 3 plus 3 plus 1 is 4.

[00:01:16]Now, watch this.

[00:01:18]Let me factorize this.

[00:01:20]So, this become 2 to the power of x minus 1 to be equal to 3 is a common factor between these two things, right?

[00:01:28]So, we shall have 3 into x minus 1, right?

[00:01:33]plus 4. Of course, 3 times x is 3 x. 3 times 1 is 3.

[00:01:39]Okay?

[00:01:40]That is plus 4.

[00:01:42]Now, look at this. This is x minus 1.

[00:01:44]This is x minus 1. So, we can say let a variable let um y to be equal to x minus 1. So, anywhere I see my my x minus 1, will input my Y.

[00:01:59]So, this become two to the power of Y to be equal to three multiplied by Y plus four.

[00:02:10]Okay? Now, let me move this three Y to the left hand side.

[00:02:15]There's something I want to do with this.

[00:02:17]So, this become this implies that two to the power of Y when I move three Y to the left hand side it becomes minus three Y.

[00:02:27]All this is equal to four.

[00:02:30]Now, watch this out.

[00:02:33]I can rewrite this four as four times one. So, we shall have two to the power of Y minus three Y to be equal to four times one.

[00:02:48]Right?

[00:02:49]So, I can further simplify this.

[00:02:52]This implies that two to the power of Y minus three Y to be equal to now, four times let me say this one can be also written as four minus one because sorry, four minus three.

[00:03:13]Okay?

[00:03:14]Four minus three is one.

[00:03:18]All right. So, let me multiply these together. So, this become two to the power of Y minus three Y to be equal to four times four is 16. So, we shall have 16 minus four times three.

[00:03:35]Okay? So, we shall have um three multiplied by four. If we multiply these together, right?

[00:03:44]>> [snorts] >> Now, let me write 16 as two to the power of four.

[00:03:49]So, this becomes two to the power of Y. I hope you are following me up to this point.

[00:03:56]Please share this video and subscribe.

[00:03:59]- 3 Y to be equal to instead of this 16 put in your 2 to the power of 4 3 * 4.

[00:04:11]Now, this is where comparison comes in.

[00:04:14]Watch this.

[00:04:16]If I equate 2 to the power of Y to 2 to the power of 4, notice that Y will be equal to 4.

[00:04:22]Okay?

[00:04:23]If I equate 3 Y to 3 * 4, that means Y will be equal to 4.

[00:04:29]So, from the comparison of this equation we we know that Y will be equal to 4.

[00:04:34]That's the value of Y. Let's find the value of X from here.

[00:04:38]Now, we notice that Y is equal to X - 1.

[00:04:43]But, our Y is 4. That means 4 will be equal to X - 1.

[00:04:50]You can collect things by moving -1 to the right hand side or left hand side.

[00:04:54]It becomes +1. So, we shall have X to be equal to 4 + 1.

[00:05:00]This implies that the value of X is equal to what? 4 + 1 is 5. That's the the value of X which is equal to 5.

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[00:05:12]Thank you.