Olympiad Mathematics | Indian | Can You Solve This New One?

Ajouté : 2026-05-28

[00:00:00]Hello everyone.

[00:00:02]If you're ready, let's provide a solution to this problem here.

[00:00:09]We have 5 to the power of B plus 5 to the power of B plus 5 to the power of B equals 2,500.

[00:00:21]So, how do we solve this?

[00:00:24]You know, we have the same thing, right?

[00:00:26]So, we can add this to get three.

[00:00:30]And um it's multiplying 5 to the power of B.

[00:00:37]And on the other hand, we have 2,500.

[00:00:43]Okay, so the next step is that we should have three into 5 to power B divided by three.

[00:00:56]Then we have 2,500 divided by three.

[00:01:01]This will take this out.

[00:01:04]And then we have 5 to the power of B to be equal to 2,500 divided by three.

[00:01:15]And if you look at this, you can't divide this without remainder, right?

[00:01:20]So, let's leave it this way and then we take the log of both sides.

[00:01:24]We have log 5 to power B to be equal to the log of 2,500 over three.

[00:01:35]Right? And then we apply a law that says log X over Y can be written as log X minus log Y.

[00:01:47]So, I'm going to apply the same thing here so that we can have log 5 to power B to be equal to log 2,500.

[00:02:01]Then we have minus log 3.

[00:02:06]Okay. Now, this 2,500 we're going to break it cuz we can have um a couple of five from there, right?

[00:02:14]Let's break it down.

[00:02:17]Um five multiply by 500 will give us 2,500, right?

[00:02:26]Then we have 5 * 5 * um how do I do it? * 100.

[00:02:35]It will still give us 2,500, right?

[00:02:39]Then from here we can have 5 * 5 * 5 * 20.

[00:02:48]Yes, this will still give us 2,500.

[00:02:52]And um for break it again, you're going to get five * 5 * 5 * 5 * 4.

[00:03:00]So, this is 2,500. As a matter of fact, 5 to the power of 4 * 4 is the same as 2,500.

[00:03:11]So, we're going to put this in place of that 2,500.

[00:03:16]So, [snorts] we have our log 5 to the power B equals log 4 squared.

[00:03:24]No, 5 to the power 4. Yes, that's what we we just got.

[00:03:28]Multiply by 4.

[00:03:31]Okay, so this is what we just got then minus log 3.

[00:03:37]And then from here, we need to do something.

[00:03:41]Yes, we need to do something especially about what we have here. So, this is log 5 to the power B mean equal to log 5 to the power of 4 plus log 4.

[00:03:56]From one of the laws of logarithm because this is multiplication, we're going to add the two logs, log 5 to the power of 4 plus log 4.

[00:04:06]Then we have our minus log 3.

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[00:04:18]So here we have log 5 to the power of B and it's equal to Um oh, I was going to do something, right? Let me remove this.

[00:04:28]So we're going to drop down the powers.

[00:04:31]This is power B, it comes down. We have B log 5. That is power 4.

[00:04:37]So we have 4 log 5.

[00:04:41]Then we have plus log 4.

[00:04:44]Then we have minus log 3.

[00:04:47]This is not that hard.

[00:04:50]You just have to know the laws of logarithm. That is all.

[00:04:53]So at this point, we're going to divide all through by log 5.

[00:05:00]Okay, we're going to divide all through by log 5. So we have this divided by log 5.

[00:05:06]This divided by log 5.

[00:05:09]Then this divided by log 5.

[00:05:13]And then by log 5.

[00:05:15]This will take this out, but this can't cancel out. So this will take this out.

[00:05:20]At the end of the day, we have B to be equal to 4 cuz there's 4 here.

[00:05:27]Plus we have log 4 to base 5. Because this from one of the laws of logarithm this 5 can become the base of the 4. So we have log 4 to base 5. Then minus the same thing here.

[00:05:43]We have log 3 to base 5.

[00:05:47]Okay, so from here we have our value of B, but it is a long one.

[00:05:54]Yes, it is a long one. Now, we are going to verify this to be sure that what we have done is correct.

[00:06:01]Okay, and it is verifying this that is even more difficult than solving it.

[00:06:09]This is the equation and our value of B is um Okay, we have four plus log four to base five minus log three to base five. So, we want to put this into the problem, but before then simplify the problem as we got three multiplied by five to the power of B to be equal to 2,500.

[00:06:39]Now, in place of um Permit me, in place of this five B, let me pick out five B.

[00:06:45]Five B is going to be you know, five to the power of four plus log four to base five then minus log three to base five.

[00:07:02]I told you verifying this may be more difficult than solving it.

[00:07:09]>> [snorts] >> But, if you know the laws, it's still very simple. So, the first thing we will do is to apply a law of logarithm to this.

[00:07:17]So, from the law, we're going to have five to power four.

[00:07:22]Okay, [snorts] multiply by five to the power of log four to base five.

[00:07:30]This is from one of the laws of logarithm.

[00:07:32]Right? This to this power times the same five to this power.

[00:07:38]Why is it multiplication? Because the power is addition.

[00:07:41]Right? So, now this is um subtraction.

[00:07:46]So, you can decide to divide. Okay, or you multiply to 5 to the power of but it will be negative this time.

[00:07:54]Negative log 3 to base 5.

[00:07:58]So, if we go on from here, we have 5 to power 4 multiplied by log to base 5 and this 5 will go from one of the laws of logarithm.

[00:08:10]So, we'll just be left with 4 over there, then multiply by from here do something.

[00:08:17]Remove the negative first, so you have 1 over everything, which is 5 to the power of log 3 to base 5.

[00:08:26]So, we have removed the negative. Then log to base 5 will still go. So, at the end of the day here we have 1 over 3.

[00:08:35]So, what do we have? We are going to have 5 to power 4 multiplied by 4 multiplied by 1 over 3.

[00:08:45]Yes, this is what we are going to have. And from what we did before, 5 to power 4 times 4 it will give us um yes, it will give us 2,500.

[00:09:00]Then multiply by 1 over 3.

[00:09:03]So, this is what we have. Now, let's go back to where we were before now. This value is for the value of 5 to power B.

[00:09:12]2,500 * 1 over 3.

[00:09:17]Okay, coming back here, this is 3 multiplied by 5 to power B is what we just got and is 2,500 multiplied by 1 over 3.

[00:09:30]That is 5 to power B alone.

[00:09:32]And since this is multiplication, this can take this out. And on the other hand, we'll have um 2,500 from here.

[00:09:44]So, this means that we are very correct to say that the value of B is 4 plus log 4 to base 5 minus log 3 to base 5. Thank you for watching. If you enjoyed yourself, subscribe for more.