Olympiad Mathematics | Russian | Can You Solve This?

本站添加: 2026-05-13

[00:00:01]Hi everyone, if you are ready, let's solve this one very quickly.

[00:00:09]We have 5 ^ x + 1 = 4 ^ x + 2.

[00:00:18]Now this is um challenging to a lot of students because they are not able to obtain the same base. So what do you do if you cannot um obtain the same base on both sides of the equation? Now follow me. Let's do this together.

[00:00:37]Um using one of the laws of indices that says m ^ a + b is equal to m ^ a * m to the power b. I'm going to write this in this form.

[00:00:55]So we have 5 to the^ of x * by the same 5 to the^ of 1 and is equal to the same thing 4 to the^ x * 4 to the^ of 2.

[00:01:11]Now to go on with what we have we are going to get 5 ^x * 5 because 5 ^ 1 is 5 and is equal to 4 to the^ x * 4 ^ 2 and that is 16.

[00:01:29]So from here what do we do? We are going to look for a way to bring x together.

[00:01:36]We have 5 ^ x 5 ^ x right I can decide to divide this by five okay by four because I want to bring 4 to^ x here 4 ^ x then divide this by 4 to the same power of x oh this cannot cancel out right yes this cannot cancel out so 4 to^ x will still be here but it will leave here so what are we now having we are 7 5^ x * 5 / 4 ^ x and this is equal to 16.

[00:02:16]Now what again do we do looking at what we have right here?

[00:02:22]Now our target now is to remove this five from here. Okay. So you can divide this divide the left hand side by five then divide the right hand side by five as well. So that this one should remove this from there and we have 5 to the^ x over 4 to the power of x over 4 to the^ of x. And on the right hand side we still have our 16 over 15.

[00:02:58]And then we are going to apply one of the laws of indices that says if you have the law says that if you have um m to the power of a divided by n to the power of a both of them are having the same powers. So this can be written as m / n both to the power of a. Okay. So this is what we're going to apply to what we have over there as we have 5 / 4 both of them to the power of x and is equal to 16 over 5. This is five 16 over 5. And now from here what do we do? We want to have the same base but that is not possible here.

[00:03:51]So we will take the log of both sides.

[00:03:55]Okay. So we'll take the log of both sides. Take the log.

[00:04:04]Okay. Take the log of both sides.

[00:04:09]Okay. So we're going to take the log of both sides. And we have log 5 / 4 to power x to be equal to the log of 16 over 5.

[00:04:26]16 over 5. So what do we do? There's a law that says the power can always go behind. And the power here is what? x.

[00:04:34]So if it goes behind it will multiply it will multiply log 5 / 4 and here we still have our log 16 over 5.

[00:04:48]Now let's apply one other law of um logarithm. The law says that log a / b is the same thing as log a minus log um log b.

[00:05:07]Okay. A log a minus um log b. This is log a / b. So this will now be written in the form of this and this will also be in the form of this. Meaning that we have x * log 5 - log 4 and this will be equal to log 16 - log 5.

[00:05:38]Okay. So this is what we have. We will not stop here because we are getting the value of x. So what do we do? Divide this by itself. Log 5 - log 4. Then divide this by the same log 5 - log 4.

[00:06:00]This and this are going to go so that our x will be alone and it's equal to log 16 - log 5 / log 5 - log 4.

[00:06:21]Now you cannot cancel the log 5 there, right? You can't because the connection between the numerator is not multiplication and that between the denominator is still not multiplication.

[00:06:35]So you can cancel anyone from there. So [snorts] let's use our calculator so that we can get our approximate value of x. So log 16 - log 5 log 16 - log 5 from calculator is giving us approximately 0 5051 approximate value and log 5 - log 4 is giving us 0.0 0 we have 969 969 right now when we divide this we'll have approximately 5.213 5.213 2 1 3. So this is the approximated value of x.

[00:07:32]The approximated value of x. But we're not going to stop here because we want to be sure of what we have done. Let's take this back to the equation and verify very quickly.

[00:07:46]Okay. If you can remember this is the value um the original equation and our value of x is approximately 5.213.

[00:07:58]So let's put this into the equation and we have 5 ^ of 5 okay 5 ^ 5 213 + 1. Then on the right hand side we will have 4 to the^ of 5.213 + 2. Now let's add what we have. So we have 5 to the^ of this + 1 will give us 6.213.

[00:08:32]Then on the right we have 4 to the^ of 7.213.

[00:08:39]Now let's get the approximated figures.

[00:08:42]down. Okay. 5 to the^ of 6.213 is approximately approximately 22,000.

[00:08:57]Yes. Now let's approximate this value too. 4 to the power of 7.213 is still approximately 22,000.

[00:09:08]Okay. So this means that our value of x which is approximately 5.213 truly satisfies the equation.

[00:09:21]Thank you for watching. If you enjoyed the way I broke it down into detail, then say something nice at the comment section and make sure you subscribe for more.