Olympiad Mathematics | This is beautifully and carefully solved

[00:00:00]Okay, so if you're ready, let's solve this one here.

[00:00:06]It looks simple, right? But I bet you you will not be able to solve it.

[00:00:11]Okay? And if you think you can, let's work this together and see the steps.

[00:00:17]This is 9 to the power of x plus 1 multiplied by 3 to the power of x all equal to 30.

[00:00:28]What do we do from here? Still on the left-hand side, we know we can write 9 as 3 squared. So, we have 3 squared.

[00:00:38]Then we have x plus 1 the power for 9.

[00:00:42]Then this is multiplying 3 to the power of x and everything equals 30.

[00:00:50]This is 3 to the power of 2x.

[00:00:55]2 * 1 is 2.

[00:00:57]Then multiplying 3 to the power of x as everything equals 30.

[00:01:05]From here now, what do we do?

[00:01:07]Remember that we are multiplying these two and we can decide to take one of the base, which is um 3.

[00:01:16]Then we have 2x plus 2 plus x.

[00:01:22]Right? Once you pick one of the base or one of the bases, you're going to add the powers.

[00:01:28]And this is equal to 30.

[00:01:32]So, if we take a step, we're going to get 3 to the power of 2x plus x.

[00:01:38]That is 3x and we have plus 2 and it's equal to 30.

[00:01:46]Now, 30 cannot be written in the base of 3.

[00:01:51]Right?

[00:01:53]Okay, this is 3 to the power 3x plus 2 being equal to 3 * 10. 30 is just 3 * 10.

[00:02:02]So, the only thing we can do right now is to take the log of both sides.

[00:02:10]And we have log 3 to the power 3x + 2 equals log 3 * 10.

[00:02:22]Okay?

[00:02:24]Now, we have log 3 to the power of 3x + 2.

[00:02:33]And it's equal to Do you know that we can split what we have here to get log 3 + log 10?

[00:02:44]Okay, if you do not know by now, I believe you know.

[00:02:48]We are adding them now because of the multiplication. If it was division here, we subtract this. We're going to subtract this.

[00:02:59]And then apply one of the laws that has to do with the power.

[00:03:04]Okay, the law that has to do with the power is um like this. Log A to the power B can be written as B log A.

[00:03:15]Right? So, this is very, very possible.

[00:03:18]So, we're going to use this to bring the power down.

[00:03:23]And then we now have 3 x + 2 to multiply log 3.

[00:03:32]Then we have log 3 + log 10 on the other side.

[00:03:40]And by the way, we know that log 10 is the same thing as 1, but we'll prefer working with the log 10 there.

[00:03:47]So, we divide this by log 3 so that this will be on the left hand side alone.

[00:03:54]Divide by log 3.

[00:03:56]Then divide this by log uh 3.

[00:04:00]This will cancel this out and this will take this out.

[00:04:04]Now, this is what we have left.

[00:04:08]>> [snorts] >> Okay, so this is what we have left after canceling out the log 3.

[00:04:14]It's only appearing here now. But to remove this log 3, we can even make 3 to be the base to 10.

[00:04:22]And we'll have 3 x + 2 equals 1 + log 10 to base 3.

[00:04:31]Okay. So, the next step is that I should take this two to the other side.

[00:04:38]And I'll have 3 x to be equal to 1 + log 10 to base 3.

[00:04:46]And then the two becomes minus two here.

[00:04:51]Now, we have 3 x to be equal to 1.

[00:04:56]Okay, the one will go because we are going to have 1 minus log We're going to have 1 minus 2.

[00:05:05]1 minus 2 is minus 1.

[00:05:08]So, this is what we have.

[00:05:10]And to continue with this, we are going to get the value of x by multiplying both sides by 1 over 3.

[00:05:21]So, we have 1 over 3 multiplying 3 x.

[00:05:25]Then on the other hand, we have 1 over 3 multiplying log 10 to base 3 minus 1.

[00:05:36]Can you see that?

[00:05:37]So, the next um three will take this three out and we have x.

[00:05:43]So, our x is now equal to 1 over 3 multiplying log 10 to base 3 minus 1.

[00:05:54]So, if we want, we stop here and this is the value of x in terms of log.

[00:06:03]Right?

[00:06:04]Now, let us go back and put this value into the equation that we have solved.

[00:06:13]The equation that we have solved is 9 to the power of x plus 1.

[00:06:19]Right?

[00:06:20]Multiplying 3 to the power of x equals 30.

[00:06:25]So, this means that if we put our value of x now, we should have 30.

[00:06:30]If you put your x now, you you're going to go a long way to prove this.

[00:06:36]So, the next thing you should do is you simplify the left-hand side.

[00:06:40]This is 3 to power 2, but the 2 will multiply x to give us 2x.

[00:06:47]The 2 will multiply 1 to give us 2.

[00:06:50]Then, this is multiplying 3 to the power of x and all of this will still be 30.

[00:06:58]Now, if you pick one of the bases, you have 3.

[00:07:01]Why are we picking one of the bases?

[00:07:03]Because we are multiplying them.

[00:07:07]Then, we add 2x plus x. That will give us 3x.

[00:07:11]Then, we have plus 2.

[00:07:14]And this is still equal to 30.

[00:07:16]So, at this point, what do you do?

[00:07:19]Put in the value of x.

[00:07:21]So, we're going to have 3 to the power of 3 into the value of x, which is 1 over Okay, let me turn this one.

[00:07:33]Okay, so we have this 1 over 3 multiplying log 5.

[00:07:43]Let me be sure of that. It is log 10 by the way.

[00:07:47]Okay, so multiplying log 10 to base 3 and we have minus 1.

[00:07:55]So, this alone is 43 x. Close this and we have plus 2.

[00:08:03]I hope you you will not be confused here.

[00:08:06]This 3 is this one.

[00:08:08]The whole of this is for the x. This plus 2 is over there.

[00:08:14]So, that to continue with this since this is this multiplying everything you can just cancel out and this will go.

[00:08:23]So, that we now have just 3 to the power of log 10 to base 3 minus 1 and we have plus 2 there.

[00:08:35]If you go on, you have 3 to the power of log 10 to base 3 then minus 1 plus 2, that will give us plus 1.

[00:08:46]Now, our duty now is to simplify this until we have 30.

[00:08:53]Okay, so let's do that.

[00:08:56]Okay, so this is where we are and um let's apply one of the laws of indices here.

[00:09:03]This is 3 to the power of log 10 to base 3 then multiply by 3 to the power of 1.

[00:09:14]I have applied multiplication because of the addition.

[00:09:17]Once you have addition between the powers you can split it by multiplying and they will have the same base.

[00:09:26]Base 3, base 3 because of this one.

[00:09:29]Then from one of the laws of indices this and this will go because they are the same. Log 3 to base 3 can cancel out. So, the only thing left here is 10.

[00:09:41]And then it's going to multiply 3 to the power 1. 3 to the power 1 is 3.

[00:09:46]And at the end of the day we have what?

[00:09:48]30.

[00:09:49]No wonder the equation that we have solved is 9 to the power of x plus 1 multiplied by 2 to the power of x it's 3 multiplied by 3 to the power of x equals 30.

[00:10:05]The same 30 that we just had.

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