Olympiad Mathematics | Indian | Can you solve this? | Verified

Ajouté : 2026-05-05

[00:00:02]Hi everyone.

[00:00:06]Let me show you how to solve this problem here.

[00:00:12]Solution.

[00:00:15]The equation is 5 to the power of x - 100 = 0.

[00:00:26]Okay, so what do we do first?

[00:00:29]Move this to the other side, right?

[00:00:32]Remember the additive inverse of -100 is +100.

[00:00:38]So, the additive inverse is what will be on the right-hand side, and that is 100.

[00:00:46]And 100 + 0 will still give us 100.

[00:00:50]Okay, so from here we look at the 100 over here.

[00:00:56]Now, can it be in the base of 5?

[00:01:00]The answer is no.

[00:01:03]But we know that 5 * 20 will give us 5 um 100.

[00:01:11]And 5 is still a factor of 20.

[00:01:15]So, 5 into 20 will give 4, right?

[00:01:19]So, 5 * 5 * 4 is 100.

[00:01:25]And at this point we say that 5 to the power of x is equal to 5 squared * by 4.

[00:01:37]So, that we will now take um the log of both sides. Since the base on the right is not equal to the base on the left completely.

[00:01:48]So, we take we take the log of both sides.

[00:01:59]log 5 to power x is equal to log 5 squared * 4.

[00:02:11]So, from here now we are going to apply some laws of logarithm.

[00:02:17]Especially to the right-hand side.

[00:02:19]There are two laws I'm going to apply to the right-hand side.

[00:02:25]Now, the first law log a b is equal to log a + log b.

[00:02:38]Okay?

[00:02:41]And then, that means that log 5 to power x will be equal to log Our a now is 5 squared, so we put 5 squared here.

[00:02:53]And our b is 4.

[00:02:56]+ log 4.

[00:02:59]Now, what do I do? I told you two laws will be applicable to the right-hand side.

[00:03:05]And one will be applied to the left.

[00:03:10]And we also know that log ab log a to power b is the same as b log a.

[00:03:19]So, this law will apply to this and also apply to this.

[00:03:24]Because both of them have powers.

[00:03:27]The x here is the power, so it goes behind to give x log 5.

[00:03:34]Then, two here is the power, so it gives us 2 log 5.

[00:03:40]Then, we have our + + log 4, right?

[00:03:46]So, we write our log 4.

[00:03:51]And um from here we are going to divide both sides of the equation by log 5.

[00:03:59]I will let you know why we are dividing by log 5.

[00:04:03]So, that this log 5 can cancel this one.

[00:04:06]So, only x remains on the left-hand side now.

[00:04:10]And our x will be two log 5 + log 4 all over log 5.

[00:04:24]Okay?

[00:04:26]And then, if we go on from here, we can distribute this um log 5.

[00:04:31]So, that it will work for both both equation um both numerators.

[00:04:37]So, let's do that.

[00:04:41]Okay, so from here now this log 5 will cancel that.

[00:04:48]So, that we have x to be equal to 2 + log 4 / log 5.

[00:05:00]Now, from here again, we can apply change of base to this.

[00:05:05]Because we know that if we have log a / log b both of them are in the same base.

[00:05:13]Now, we can change the base and this will be equal to log a to base b.

[00:05:21]Okay, so I will apply the same thing to this.

[00:05:24]So, that we have x to be two + log 4 to base 5.

[00:05:32]Okay, so the 5 becomes the base.

[00:05:36]And this right now is our value of x.

[00:05:41]We want to put this back into the equation, which is 5 to the power of x - 100 = 0.

[00:05:49]So, this means that 5 to the power of x here is expected to be 100.

[00:05:56]Because it is 100 - 100 that should be 0.

[00:06:00]So, let's come down to 5 to the power of x.

[00:06:05]5 to the power of x will now be 5 to the power of 2 + log 4 to base 5.

[00:06:16]So, that the um our x is now replaced over here.

[00:06:24]And then, if you go on again, we can write this as 5 squared * 5 to the power of log 4.

[00:06:35]Both to base 5.

[00:06:37]Okay, so this is what we have.

[00:06:40]And there is a law that says Let me write it here. There's a law again that says a to the power of log b to base a is equal to b.

[00:06:55]Right? This is one of the laws of logarithm.

[00:06:59]So, if this is true, then we can relate it to what we have over here.

[00:07:04]So, that we have 5 to the power of 2 * by the whole of this now will be 4.

[00:07:10]Comparing it to what we have here.

[00:07:13]And 5 squared is 25 * which is 100.

[00:07:21]So, that means that we are very correct to say the value of x is equal to 2 + log 4 to base 5.

[00:07:34]Thank you for watching.

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