Olympiad Mathematics | Russian | Can You Solve This?

Added: 2026-05-05

[00:00:00]Hi everyone.

[00:00:04]Let's let me show you how to solve this problem very quickly.

[00:00:12]Okay, we have two to the power of X plus three equals three to the power of X plus one.

[00:00:23]Now, what do we do first?

[00:00:26]Do you remember this law of indices?

[00:00:29]M to the power of A plus B that this can be written as M to power A times M to power of B.

[00:00:40]So, if this is true, then what we have on both sides of the equation will be in this form.

[00:00:48]And we will have two to the power of X multiplied by the same two to the power of three.

[00:00:58]Now, this is equal to three to the power of X multiplied by the same three to the power of one.

[00:01:08]So, what's are we saying?

[00:01:10]We are having two to the power of X multiplied by eight because two to power three is eight.

[00:01:18]And this is equal to three to the power of X multiplied by three because three to the power of one is the same as three.

[00:01:28]Now, I'm thinking what we're going to do next.

[00:01:32]Um here is what we are going to do very quickly. We're going to divide both sides of the equation by eight.

[00:01:41]Okay, so that these can go.

[00:01:44]And we'll be having um two to the power of X to be equal to three to the power of X multiplied by three over eight.

[00:01:59]Okay. So, the next thing we're going to do is to remove or bring the terms with X to the same sides of the equation.

[00:02:07]So, we don't take these to this side.

[00:02:10]Rather, we take these to this side so that three over eight will be alone and this will work with this.

[00:02:16]So, we're going to divide this by three to the power of X.

[00:02:21]Then divide the left-hand side by three to the power of X. Three to the power of X is canceling each other.

[00:02:29]And we'll have two to the power of X.

[00:02:34]Okay, two to the power of X over three to the power of X and is equal to three over eight.

[00:02:45]Now, we can compare or better still combine what we have here.

[00:02:51]The [snorts] numerator has power of X and the denominator has power of X.

[00:02:57]So, the combination now will give us two over three to the power of X to be equal to three over eight.

[00:03:11]Okay. And now, because the bases are not the same we are expected to take the log of both sides.

[00:03:20]So, that means we have the log of two over three to the power of X to be equal to the log of three over eight. Let's close this.

[00:03:34]Okay, so let's continue from here.

[00:03:39]Okay, so from here the power X will go behind and we'll have log three over Okay, that's supposed to be two over three.

[00:03:53]So, we have log two over three.

[00:03:57]And that will be equal to log three over eight on the other side, right?

[00:04:04]And then there's another law again that we can apply over here.

[00:04:08]If we have log C over D you know, this can be written as log C minus log D from one of the laws of logarithm.

[00:04:22]So, this means that we are having um X multiplying log two minus log three.

[00:04:35]Then on the right-hand side, we have log three minus log eight.

[00:04:41]By the way, this is subtraction, right?

[00:04:45]Okay, so we continue with this.

[00:04:48]We want to find the value of X.

[00:04:53]So, there'll be need for us to divide both sides by this.

[00:04:57]So, we have log two minus log three.

[00:05:03]Then the same thing happens there.

[00:05:05]Log two minus log three.

[00:05:09]So, that this one can cancel this out.

[00:05:15]Okay, so the value of X is log three minus log eight.

[00:05:26]Okay, log three minus log eight all over log two minus log three.

[00:05:35]Okay, let me show that.

[00:05:37]So, this is what it is.

[00:05:40]And from our calculator, log three minus log eight will give us negative 0.4 the negative 0.426 approximately.

[00:05:54]And then log two minus log three will give negative 0.

[00:06:01]Okay, let me confirm that. 0.176 176.

[00:06:09]Okay, so this is what we have.

[00:06:12]And the division here will give us X to be equal to 2.42.

[00:06:20]In fact, approximately 2.42.

[00:06:24]So, this is the approximated value of X.

[00:06:29]Now, let's put this back into the equation very quickly to see if we're going to have the you know, the value of X mean the value on the left on the left-hand side and that on the right-hand side.

[00:06:44]Let's see if the two values can be approximated you know, can be approximately equal to each other.

[00:06:54]Okay, so this is the equation.

[00:06:57]And our X is 2.4 two, right? So, that means we're having two to the power of 2.42 plus three.

[00:07:08]Now, on the right-hand side, we have three to the power of two 2.42 then plus one.

[00:07:17]Now, on the left, we have three two to the power of five.42.

[00:07:26]That is 2.42 plus three.

[00:07:28]Then on the right-hand side, we have three to the power of 3.42.

[00:07:35]Now, let's use calculator and see what we are going to have.

[00:07:40]On the left-hand side two to the power of 5.2 two to the power of 5.42 is 42.8.

[00:07:53]Okay, and this continues, right?

[00:07:56]Then on the other hand we're going to have three to the power of four three to the power of 3.42 and that is 48.

[00:08:08]48. Okay, sorry, 42.

[00:08:12]42.8 as well.

[00:08:16]Okay, so this means that our value of X is satisfying.

[00:08:22]So, X to be equal to 2.42 satisfies the equation.

[00:08:28]Thank you for watching. If you enjoyed yourself subscribe so that I can we can have more videos like this.