Olympiad Mathematics | Indian | Can You Solve This?

追加: 2026-05-07

[00:00:01]If you are ready, let's solve this one.

[00:00:06]Solution, we have 4 to the^ of a * 4 to the^ a equ= 80.

[00:00:17]Okay. So, how do we solve this? Here we have the same base. We have the same power. So, how do we work it? It's very simple. Now there are two ways you can work on the left hand side. It's either you pick one of the bases, then you add the powers, you know, a + a, right?

[00:00:40]Or or I'm still working on the left hand side, by the way. You multiply the bases and then pick one of the powers, right? So the two of them will give us the same value. So to continue with this we have our 4 to the^ of a + a that will give 2 a right and it's equal to 80.

[00:01:08]Yes it's equal to 80 on the other side.

[00:01:12]And if you look at 80 80 cannot be in the base of four completely. So there will be need for us to break that um 80 so that we'll have um 4 * 20 4 * 20 is 80 right okay so 4 to the^ 2 a will be equal to 4 * 20 is 4 * 5 right so 4 * 4 * 5 is equal to 80. And from here we can have 4 ^ 2 a to be equal to 4 ^ 2 * by 5.

[00:02:01]Now we we don't have the same base completely. So there'll be need for us to take the log of both sides. And from here we have log 4 ^ 2 a and it's equal to log 4 ^ 2 * 5.

[00:02:23]Now let's apply some laws.

[00:02:27]Remember that if you have log mn, we can express this as log n log m + log n.

[00:02:38]Okay. So if we compare this to this now our m is going to be 4^ 2 and our n is 5. So from the left we have log 4 ^ 2 a and is equal to log 4^ 2 + log what log 5.

[00:03:03]Okay so this is interesting. Now we have um dealt with that. The next thing is to deal with the powers. Here we have 2 a as a power and we have to also a power.

[00:03:17]So there's a law that says the powers can be brought back. Okay? Or can be brought behind. So we're going to bring 2 a behind. Then we have our log four.

[00:03:31]Bring two behind and then you have your log four.

[00:03:35]then plus log what 5 because this does not have um a known power.

[00:03:44]So what do we do? We want to um how do I put it? We are looking for the value of a right. So that means I have to divide both sides divide all through by log four so that I'll be left with 2 a first then I will know what to do to the two.

[00:04:07]Okay. So we are going to divide this by log four.

[00:04:13]Then divide the whole of this by log four. log four will cancel itself so that 2 a will be equal to what we have here. But mind you, we can split what we have here too to get 2 log 4 / log 4 then + log 5 / log 4 so that this one can cancel this as well. So our 2 a will be equal to 2 + log 5 / log 4.

[00:04:58]And without wasting time we will apply change of base to this part alone. So that we will now have 2 a to be equal to 2 + log 5 to base 4. The four here becomes the base of the five. Now we're not stopping here because we are getting the value of um a. So I need to remove two from here.

[00:05:25]Meaning that I divide both sides of the equation by two. So we divide this by two and divide the whole of this by two.

[00:05:33]Now two will go again with two. And our a is equal to half of everything that we have here which is 2 + um log 5 to base 4.

[00:05:51]Yes. Because when you multiply half by a number you're dividing that number by two. And to work this we can open the bracket so that our a will be equal to 1 / 2 * 2 is 1 plus here we have half of log 5 to base 4.

[00:06:13]Yes half of log 5 to base 4. As a matter of fact this is our value of a. Yes, this is our value of a and we are going to confirm it because I know you might be asking what if this is not correct.

[00:06:29]You're going to see for yourself as we try to verify. Let's go back.

[00:06:36]Okay, so this is the equation before us and our value of a is 1 + 1 / 2 log 5 to base 4. So, how do we how do we um verify this? Now, we're going to have four to the^ of 1 + 1 / 2 log 5 to base 4, right?

[00:07:05]And I'm having this into in two um places. Yes, we're going to have this in two places. Meaning that we can just square whatever we have here. think it's faster that way. Yes. Whatever we have here, we are going to square it and see if it will give us 80. From here, we apply um some laws. Okay, we're going to apply one of the laws of indices to this. This is the same as 4 to the^ of 1 * the same 4 to the^ of 1 / 2 log 5 to base 4.

[00:07:46]Okay. And everything here is still squared.

[00:07:51]Now in case you don't know what I've done, since we are multiplying these two, you pick one of the bases to get to get four. Then you add the two powers to give you the power over there.

[00:08:03]Right? Then the next thing you're going to do, remember that, let me explain this very quickly, that if you have um let's say you have a let me use a real number. Let's say you have two log five to base four, right? You have 2 log 5 to base 4. This can be expressed as log 5^ 2 to base 4. This two here is a power to the 5. Right?

[00:08:37]Okay. So, it's the same thing that I'm going to do to this one. So we will now have 4 ^ 1 * 4 ^ 1 * 4 again to the power of log 5 to power 1 / 2 okay log 5 to the^ 1 / 2 but to base 4 this 1 /2 here is a power to the five right and mind you 5 to the^ 1 /2 is the same thing as the square root of 5. So let's take that step. Whatever that we do we will still square it because of that. So to go on from here 4 to^ 1 is already four. So let's use four. Then multiply by 4 to the power of log square roo<unk> of 5 to base 4.

[00:09:37]Yes. 5 ^ 1 /2 is a square root of 5.

[00:09:41]Everything is still squared.

[00:09:44]And then from here you um you realize that the base here is still the base to the log. And there's a law that says log to base four can cancel out this four.

[00:09:56]So at the end of the day we have four *<unk> 5 and then everything is squared.

[00:10:06]Interesting, right? By the way, we are trying to see if we're going to have 80.

[00:10:11]So, if we don't have 80, then we are not correct. So, this implies that we have we can combine this so that it will be under the same square root sign. Yes, we can do that. So, that this will be under the same square root uh sign. And to do that, we will square this one first.

[00:10:32]We're going to square this 16, right? So 16 * 5. So that will give us the square root of 80.

[00:10:42]Remember there's a square over there.

[00:10:46]There's a square over there. Okay. I hope you can see the square that is there. There's a square over there.

[00:10:54]Right? So now we are having the square root of 80 all squared which is equal to 80 cuz square and square roots can go.

[00:11:04]So at the end of the day we are very correct to say that our a is equal to 1 + 1 / 2 log 5 to base 4.

[00:11:21]Thank you for watching.