Indian | Olympiad Mathematics | Can You Solve This One?

Added: 2026-05-30

[00:00:01]Let's solve this problem here.

[00:00:04]This is 4 to power x + 4 to power x = 16 / 3.

[00:00:12]16 / 3 is going to give us decimal value, right?

[00:00:19]So, [snorts] we will leave it the way it is.

[00:00:21]And then we'll work on the right-hand side.

[00:00:24]4 to power x + 4 to power x will give us two of four to power x.

[00:00:31]And this is equal to 16 over 3.

[00:00:35]You know, we won't stop here, right?

[00:00:37]We have to remove this two from here.

[00:00:39]So, to do that, we're going to multiply both sides of this equation by 1 over 2.

[00:00:45]So, we have 1 over 2 1 over 2 multiplying 2 Okay, so we have 1 over 2 multiplying 2 * 4 to power x.

[00:01:01]And then we have 1 over 2 multiplying 16 over 3.

[00:01:08]So, this From here now, what do we do?

[00:01:12]This can go into that, so we have 4 to the power of x.

[00:01:16]So, we're going to write 4 to power x on the left. Then on the right the right-hand side, what do we have? We're going to have um 1 over 2 * If we multiply 1 over 2 by 16 over 3, we're going to have um 8 over 3.

[00:01:37]How did we get 8?

[00:01:40]Because we know that 2 into 16 is 8, and this three will still come down.

[00:01:46]So, from here we take the log of both sides.

[00:01:50]We have log 4 to power x to be equal to log 8 over 3.

[00:01:58]Okay, I hope you know your laws of logarithm.

[00:02:02]Okay, if you do Okay, in what other way can you express log 8 over B?

[00:02:11]If you know your laws of logarithm.

[00:02:14]Okay.

[00:02:15]log A minus log B from the laws of logarithm. So, I will express this like this.

[00:02:26]So, I'm going to get um log 4 to the power X to be equal to log 8 minus log um 3.

[00:02:40]But then, look at 8. We can break 8, right?

[00:02:43]So that we get log 4 to the power X to be equal to log 4 * 2 then minus log 3.

[00:02:53]4 * 2 is 8. Then, we can express this again in a different way so that we have log 4 to the power X being equal to log 4 plus log 2.

[00:03:09]This is coming out of this, right?

[00:03:11]Because of the multiplication.

[00:03:14]Then, we have our minus log 3.

[00:03:18]So that at this point we can now divide No, don't divide yet.

[00:03:23]We don't divide yet because of the power. So, bring this power behind, so we have X log 4 being equal to log 4 plus log 2 minus log 3.

[00:03:38]>> [snorts] >> So, at this point, we can now divide all through by log 4.

[00:03:44]That will give us log uh will give us X on the left-hand side.

[00:03:52]Okay, so we're going to divide all through by log 4.

[00:03:58]Divide this by log 4.

[00:04:01]Divide this by log 4.

[00:04:03]Then we divide this by log 4.

[00:04:08]Okay. So, this can take this out.

[00:04:13]Our X will be equal to this into this is 1 plus there we have log 2 divided by log 4.

[00:04:24]Then minus log 3.

[00:04:28]Okay, log 3 divided by log 4.

[00:04:34]So, from here we are getting our value of X. But we can still simplify what we have here.

[00:04:41]Yes, we can simplify. You know what?

[00:04:43]Let's just do it this way. So, X will now be 1 plus 4 becomes the base to 2.

[00:04:49]So, we have 2 to base 4.

[00:04:53]Then here we have minus log 3 to base 4. So, let this be our value of X.

[00:05:03]Although we can still simplify this further, but let's leave it this way.

[00:05:07]Then we bring down the equation that we have solved which is 4 to the power X plus 4 to the power X being equal to 16 over 3.

[00:05:19]So, we'll now focus on the left-hand side.

[00:05:23]This is 2 of 4 to the power of X.

[00:05:29]Right?

[00:05:30]And if we pick out 4 to power X it will imply that we have 4 to the power of 1. That is the power of X now, the value of X.

[00:05:43]Plus log 2 to base four minus log three to base four.

[00:05:52]So, from here this is four to the power one multiplied by four to the power of log two to base four.

[00:06:04]Mine Okay, we have that, right? So, we multiply it by the same four, but to negative power. So, this is negative log three to base four.

[00:06:18]Then we apply a law.

[00:06:22]This and this are going to go. Four to the power of log two to the same base is going to go, so two remains.

[00:06:29]Four to the power of one is four times two.

[00:06:33]Four times two, then times from here.

[00:06:37]Remove the negative first as we have one over everything.

[00:06:42]And that is um four to the power of log three to base four.

[00:06:54]Okay. So, the negative has gone. Then, the law that I explained earlier is also reflected here.

[00:07:01]Four to the power This is base or this is the base.

[00:07:05]And this is the exponent. So, four to the power of log three to the same base of four can always go. So, that this remains three.

[00:07:14]So, at the end of the day we have four times two times one over three. This value This value here is for four to the power X alone.

[00:07:26]So, let's go up.

[00:07:28]Okay. So, from here, remember that our four to the power X that we just got is four times two times one over three, right?

[00:07:39]And we say that on the left-hand side this two is the same as two of four to power of X.

[00:07:48]So, [snorts] this implies that we're having two times four times two times one over three. So, this is what it means.

[00:07:59]Are we going to get um 16 over three?

[00:08:03]That's what we want to be sure of.

[00:08:06]Now, 8 * 2 4 * 2 is 8. 8 * 2 is 16.

[00:08:10]And it's 16 over three. Look at it over there.

[00:08:14]So, this means that our equation, our value of X is correct. Um the value of X is truly 1 plus we have log two to base four minus log three to base four.

[00:08:29]Thank you for watching.

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