Olympiad Mathematics | Indian | Can You Solve This One?

Added: 2026-05-12

[00:00:01]In one of my previous videos, I solved a problem like this, and some people found it really confusing.

[00:00:11]>> [snorts] >> Let's go through this one, so that you'll be able to get all the steps.

[00:00:18]Let's go. We have 3 ^ a + 2 = 4 ^ a - 1.

[00:00:26]Now, the only thing that will make this confusing or challenging is the fact that the bases cannot be the same.

[00:00:34]Yes. There's no manipulation that will make you get the same base on both sides of the equation. So, what do you do?

[00:00:42]You have to obey some laws of indices until you can bring your a to the same side, and then take the log.

[00:00:52]But, look at this.

[00:00:54]Imagine that you have m to the power of x + y.

[00:01:00]You know, it's already in the form of what we have here.

[00:01:03]This is the same thing as m to the power of x * m to the power of y.

[00:01:11]The same thing is applicable to this, but the other one is negative. Right?

[00:01:15]So, let's go with this.

[00:01:18]Let's go with this. We're going to have our 3 ^ a now multiplied by the same 3 ^ 2 on the left, and on the right-hand side, we have 4 ^ a multiplied by 4 ^ -1.

[00:01:37]Interesting, right?

[00:01:38]Let's take a step, so that we can get 3 ^ a multiplied by 9.

[00:01:45]3 squared is 9. Then, here we have 4 ^ a multiplied by 1 over 4.

[00:01:53]1 over 4 is the same thing as 4 ^ -1.

[00:01:57]And that is from one of the laws.

[00:01:59]Yes, that's from one of the laws of indices.

[00:02:04]So, what do we do?

[00:02:06]In this case, let's bring the terms with A together.

[00:02:11]So, I want to divide this by 3A 3 to the power A rather, then divide the whole of this by 3 to the power of A.

[00:02:20]So, if I do that, I'm going to get A and 3A 3 to the power A to cancel this 3 to the power A.

[00:02:28]So, what is left on the left-hand side is nine.

[00:02:34]Okay, what is left on the left is nine.

[00:02:38]Sounds somehow.

[00:02:40]Then, here we're going to have four to the power of A divided by 3 to the power of A, right?

[00:02:50]Okay, so this is what we are having. And then we multiply by 1/4. But, our next target is to divide both sides of the equation by 1/4 so that this one here will come to meet with this.

[00:03:06]Yes, or better still, if you don't want to divide, what do you do? You cross multiply because we know that this is over one. Four will multiply nine >> [snorts] >> and it will leave here.

[00:03:18]Yes, that's very, very possible. So, 4 * 9 is on the left and it's equal to here we have 4 to the power A divided by 3 to the power A. If you multiply by one, it will give you the same thing.

[00:03:34]But, permit me to write the one with the variable first before the constant.

[00:03:42]So, we're getting 4 to the power A over 3 to the power A to be equal to 4 * 9 and it's 36.

[00:03:52]So, at this point, what do you think we can do?

[00:03:56]We will now apply a law. Our target is to get A in one place.

[00:04:02]That's our target from the start. So, now we're going to combine this because there's a law like this.

[00:04:09]Um M to the power A, okay, let me not use A. M to the power B over N to the power B is the same thing as M over N both to the same power of B.

[00:04:25]Now, I will do the same thing here so that 4 over 3 will have the same power of A.

[00:04:32]And this is equal to 36.

[00:04:36]So, we have succeeded in obtaining A in one place. But this time around, it is a power. And we can't have the same base on both sides of the equation. So, the next step is that we take the log of both sides.

[00:04:53]Yes, that is the only step you can take so that we have log 4 over 3 to power A to be equal to the log of 36.

[00:05:08]Yes, we have log 36. Now, the power here is going to go behind.

[00:05:13]And if that happens, if that happens, then we have A to multiply log 4 over 3.

[00:05:25]And then on the other hand, we have log 30 log 36.

[00:05:31]Okay, so let's go on.

[00:05:35]Okay, so we're going to continue from here.

[00:05:39]Um remember we're trying to get the value of A.

[00:05:42]But we can even apply a law here because we know that log x over y is the same as log x minus log y.

[00:05:54]So let's apply that there and we have a to multiply log 4 minus log 3.

[00:06:04]This is equal to log 36.

[00:06:09]Okay?

[00:06:10]So we go on so that we can divide this by log 4 minus log 3.

[00:06:20]Then divide the other side by log 4 minus log 3.

[00:06:28]The whole of these are going to go.

[00:06:31]Yes, they are going to go and our a will be equal to log 36 divided by log 4 minus log 3.

[00:06:47]Yes, we have log 36 divided by log 4 minus log 3 and at this point let's use calculator so that we can get the approximated value of a.

[00:07:01]So we press calculator to get log 36.

[00:07:08]Log 36 is approximately 1.556.

[00:07:13]And that will be over log 4 minus log 3 from calculator and it's giving us It is giving us 0.12 49 approximately.

[00:07:31]So from calculator again we divide this and a will be approximately 12.46.

[00:07:42]So this is our value of A.

[00:07:45]And we have to be very sure of what we have done.

[00:07:50]As we try to verify this very quickly.

[00:07:57]Okay, so let's verify what we have done.

[00:08:00]This is the original equation and our A is equal to 12.46.

[00:08:07]So, that's on the left are going to have three to the power of 12 point four six, then plus two.

[00:08:14]Then on the right hand side, we shall have four to the power of 12.46.

[00:08:21]Then we take one away as we have in the equation.

[00:08:25]So, um on the left hand side, we have three to the power of 14 point four six addition.

[00:08:33]Then here we have um four to the power of 11.46 with subtracted.

[00:08:42]So, what do we now have from here? Let's use our calculator to be very sure.

[00:08:48]Now, from calculator approximately three to the power of 14.46 is eight million.

[00:08:57]Approximately eight million.

[00:09:01]Yes, you can confirm it.

[00:09:04]Okay, so on the other hand four to the power of 11.46 is still approximately eight million.

[00:09:16]So, what are we saying?

[00:09:18]We are saying that the value of A approximately equal to 12.46 satisfies the equation.

[00:09:29]Thank you for watching. If you enjoyed yourself, subscribe for more and let me have nice comments from you.