Math Olympiad | Indian | Can You Solve This One?

Added: 2026-05-20

[00:00:01]Hi everyone.

[00:00:05]Are you ready? If you are, let's solve this problem here.

[00:00:10]4 to the^ a * 4 to the^ a -1 = 20.

[00:00:19]Now I want you to you know pay attention as I explain everything into detail.

[00:00:28]So what do we do first?

[00:00:30]we work on the middle term 4 to ^ a - um 1. This is because imagine that we have something like this m the^ um n minus one or use another letter here n minus p right this can be written as m to the^ n * m to the^ of p okay the same thing to the power of p so I'll explain the middle term in this form we have 4 to power a now then multiply by 4 to the power of a * the same 4 to the power of -1 and all of this is equal to 20.

[00:01:21]So the next point is that we apply one of the laws of indices that says that if you're multiplying and you have the same base, you pick one of the bases and then add the powers. So I'm going to do the same thing to all of this. We pick four because we have the same base. Then we have a + a + -1, but that will still be minus1.

[00:01:48]So all of this is equal to 20.

[00:01:52]I told you I was going to show everything into detail. And by the way, if you are enjoying yourself, then you should subscribe to my channel.

[00:02:01]Here we have 4 to the^ of 2 a - 1 and is equal to 20.

[00:02:08]Now what again do we do? If we are able to express 20 in the base of four, that would have been good. But we not be able to do that. So what do we now do? We are going to work on this again. The same law I explained earlier. So we're going to get 4 to the^ 2 a * the same 4 to the^ of -1 and this is equal to 20.

[00:02:39]So the next point is that we apply a law here. So we're going to get 4 ^ 2 a * 4 ^ -1 is 1 / 4 and this is equal to 20. And at this point we will cross multiply.

[00:03:00]So that 4 will multiply 20 and we have 4 to the^ 2 a to be equal to 4 * 20.

[00:03:11]If you go on, you're going to get your 4 to the^ 2 a to be equal to 4 * 4 * 5 because 20 is 4 * 5.

[00:03:24]Now to continue with this, we're going to have 4 ^ 2 a 4 ^ 2 a to be equal to 4^ 2 * 5. So we're thinking we'll have the same base completely on both sides. But since it is not possible, we have to take the log of both sides. And we now have log 4 to the^ 2 a to be equal to log 4^ 2 * 5.

[00:04:00]Now look at the law that I um apply I'm going to apply to the right.

[00:04:07]Okay. So here is the log that I was talking about.

[00:04:12]Do you know that log a is the same as log a? Okay, we have a in the problem.

[00:04:20]Let me not use a. Let's say log x y.

[00:04:26]log xy is the same thing as log x + log y.

[00:04:32]Right? If it is product multiplication, then it's going to be addition.

[00:04:38]If it was division, this would have been subtraction. So we now have log 4 ^ 2 a [snorts] to be equal to log 4^ 2 + log 5.

[00:04:53]Right? So this is it. And from here now what do we do to the powers? We're going to bring the powers behind following one of the laws of indices. So we are going to write 2 a then log 4 to be equal to 2 log 4 + log 5.

[00:05:15]The two powers have been shifted behind.

[00:05:20]So that we can now divide both sides of the equation by log four or better still we divide all through by log four.

[00:05:30]Okay, [snorts] this is going to go with this one and 2 a is alone on the left which is equal to this. We take this out. So we have 2 + log 5 / log 4 and we can easily apply one of the laws um change of base. See that four becomes the base to 5 and 2 a is equ= to 2 + log 5 but to base four right but we're trying to get the value of a so I'm going to divide all through by a again I mean divide all through by two divide by two divide by two okay um I'll multiply this by I'll multiply this by 1 / two. You know that's the same as dividing by two, right? So this will go, this will go and we have 1. So a is equal to we have 1 + let's write this one first.

[00:06:41]We have 1 / 2 * log 5 / 4. And from one of the laws of indices, we know that this 1 /2 is a power to five. Okay? So our a can better be written as 1 + if you take 1 /2 up there it becomes power and that means that we have the square root of 5 because 5 to the^ 1 /2 is the same thing as the square root of 5 and this is to base 4. So at this point we have our value of a but we're not stopping here. We have to verify what we have done. Let's go back to the original equation.

[00:07:26]Okay, so this is the original equation and our value of a is 1 + log<unk> 5 to base 4. Before we put in this value of a, let's work on this equation. Now this equation can give us 4 ^ 2 a - 1 = 20. Both of them are the same. So now let's put in our value of a. So we're going to get 4 to the^ 2 * a which is 1 + log square<unk> of 2<unk> 5 rather<unk> 5 to this 4 then we have - one. So if we simplify this and it does not give 20 then we made a mistake.

[00:08:20]in the video.

[00:08:23]So let's open the bracket. We have 4 to the^ of 2 * 1 is 2 plus here 2. We multiply this. So we have 2 log square<unk> of 5 to base 4. Then this is minus one. We've opened the bracket already.

[00:08:40]But mind you that these two here can cancel the square. You know the two can cancel the square root because these two can be a power to roo<unk> 5 and square root and square can cancel each other.

[00:08:56]So here now we have 4 to the power of 2 - 1 that is 1 then + log 5 now no longer a square root to base 4. Then we apply one of one of the laws of indices 4 ^ 1 * 4 to the^ of log 5 to base four log five to base four um log log to base 4 and this four can go right. So that we have 4 ^ 1 which is equal to 4 and is multiplying five. At the end of the day, we have 20 and it is 20 we had on the other side of the equation.

[00:09:40]Therefore, okay, we are solving for a. Therefore, a = um 1 + log<unk> 5 to base 4 truly satisfies the equation. Thank you for watching and thank you for subscribing.