Indian | Olympiad Mathematics | Can you solve this?

追加: 2026-05-05

[00:00:00]Hi, everyone.

[00:00:05]Let's provide a complete solution to this problem right here.

[00:00:11]2 to the power of X over 3 is equal to 12.

[00:00:19]Okay, so let's work this.

[00:00:22]The first step is to cross multiply, right?

[00:00:27]Okay, so we will have 2 X I mean 2 to the power of X to be 12 multiplied by 3.

[00:00:38]And um let's break it down.

[00:00:42]This is 2 to the power of X on the left.

[00:00:45]And on the right, we have 2 times 2 times 3, right?

[00:00:53]That is 12.

[00:00:54]Then multiplied by 3 again.

[00:00:58]Okay?

[00:00:59]2 times 2 times 3, that is 12, then times 3.

[00:01:04]So what are we saying? We are having um 2 to the power of X equals 2 to the power of 2 multiplied by 3 times 3 is 9.

[00:01:17]Okay?

[00:01:18]So we cannot have the same base on both sides of the equation.

[00:01:24]So what then do we do?

[00:01:26]We'll take the log of both sides.

[00:01:30]Okay, so we now have the log of 2 to the power of X to be equal to the log of 2 to the power of 2 times 9.

[00:01:43]And then you will look at this. This is multiplication here.

[00:01:47]And if you have log AB, this is A times B. So it is the same thing as log A plus log B.

[00:01:58]So we apply the same thing over there.

[00:02:01]So we have our log 2 to the power of X to be equal to the log of 2 to the power of 2 plus the log of 9.

[00:02:14]Okay, so this is what we have. And then this power now will go behind. So we have X log 2 to be equal to this power again will go behind as we have 2 log 2.

[00:02:31]Then we have plus log 9.

[00:02:38]From here now, what do we do? We are looking for the value of X. So we want to cancel out this log 2 bar.

[00:02:45]So that means we divide this by log 2.

[00:02:49]And then the other side, do the same thing. You divide by log 2.

[00:02:54]Log 2 into log 2 is 1. 1 times X is X.

[00:02:59]So we have that X on the other side.

[00:03:02]Then here we have it 2 log 2 plus log 9 divided by log 2.

[00:03:14]Now, what again should we do?

[00:03:17]If you stop here, you are okay, or you use calculator, you are good.

[00:03:21]But you can still simplify this.

[00:03:24]So that even if you want to use calculator, it will be easier for you.

[00:03:29]So this is X to be equal to this right now can be written as 2 log 2 over log 2.

[00:03:42]Okay?

[00:03:44]2 log 2 over log 2, then plus here we have log 9 divided by log 2.

[00:03:53]So that this one can take this out.

[00:03:57]So that we will now have um X to be 2 plus log 9 over log 2.

[00:04:09]Log 9 over log 2, right?

[00:04:12]Then from here we apply change of base so that this fraction can go.

[00:04:20]Okay, so from here we apply change of base, right?

[00:04:24]So X will now be 2 plus log 9 to base 2.

[00:04:31]This 2 here becomes the base to the 9.

[00:04:36]So we have our value of X. If I want, I can press calculator and I'll have my answer in decimal form.

[00:04:45]But then let's put this back into the equation.

[00:04:49]Remember the equation is 2 to the power of X over 3 equals 12.

[00:04:57]Okay, before we verify, let's cross multiply.

[00:05:01]So we have 2 to the power of X now to be equal to 3 multiplied by 12.

[00:05:08]And what is 3 times 12?

[00:05:12]2 to the power of X will be equal to 36.

[00:05:17]Now, let's put in the value of X and see if 2 to the power of X is going to be equal to 36 from our calculation.

[00:05:28]We have 2 to the power of X which is 2 plus we have log 9 to base 2.

[00:05:39]Okay, so apply one of the laws of indices that if you have M to the power of C plus D you can write this as M to the power of C times M to the power of D.

[00:05:56]So that means what we have here now can be in this form.

[00:06:01]So that we will now have 2 to the power of 2 right?

[00:06:09]Okay, so we have 2 to the power of 2 multiplied by the same 2 to the other power which is log log 9 to base 2.

[00:06:21]Right?

[00:06:22]So we have it 4 multiplied by this which is 2 to the power of log 9 to base 2. But we have to understand something very quickly.

[00:06:35]Um you should be able to know that if you have if you have M okay, small letter M to the power of log N to base small letter M.

[00:06:51]So the whole of this will be equal to N.

[00:06:55]Because log to base M and this M will go.

[00:06:58]That's one of the laws of logarithm as well.

[00:07:01]So from here now, we will say that the whole of this, comparing to this law, will be equal to 9.

[00:07:10]So here we have 4 multiplied by the whole of this, we have 9.

[00:07:16]Okay, and what is 4 times 9?

[00:07:19]4 times 9 is equal to 36, right?

[00:07:24]And that's what we [clears throat] had over there. Look at that.

[00:07:28]We said that 2 to the power of X is equal to 36.

[00:07:32]And this is to say that our X to be equal to 2 [snorts] plus log 9 to base 2 truly satisfies the equation.

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