Olympiad Mathematics | Japanese | Can you solve this one?

[00:00:00]Hi, everyone.

[00:00:03]Let's um provide the complete solution to this problem here.

[00:00:10]We have 2 ^ x * 3 ^ x = Okay, it's -108 = 0.

[00:00:22]In fact, the first thing you should do is to take this to the other side first.

[00:00:27]So, we have 2 ^ x * 3 ^ x to be equal to 0 + 108, and that is 108.

[00:00:38]and 8.

[00:00:41]Now, the next step is that we work on what we have here.

[00:00:47]Do you know that if we have um if you have something like this, a to the power of b * c to the power of b, this is the same thing as a * c, both raised to the power of b.

[00:01:05]Okay? So, we're going to apply this to what we have on the left-hand side.

[00:01:11]So, we will now have um 2 * 3 to the power of x, and it's equal to 108.

[00:01:22]So, that if we continue, we can multiply this to get 6 ^ x = 108.

[00:01:31]Now, looking at what we have, do we think 108 can be written in base 6?

[00:01:38]The answer is no, right? So, let's try to break it down.

[00:01:43]6 is a factor.

[00:01:45]Okay? So, 6 into 10 is going to be 1, remainder 4, making this one 48. 6 into 48 will give us 8.

[00:01:56]Okay? 6 can still go into 18. So, we have six to power X to be six multiplied by 18 is six times three.

[00:02:04]Okay, so we are good, right? So, this means that six to power X will give us um 36, which is the same thing as six to power two, then multiply by three.

[00:02:16]So, now that we we are not having the same base completely, so we have to take the log of both sides.

[00:02:25]As we have log six to the power of X to be equal to log six squared multiplied by three.

[00:02:34]Do you see that?

[00:02:36]Then the next law we are going to apply will be directed to the the right hand side because we have addition here. I mean multiplication, which will turn eventually to addition because we know that Look at this. If you have um let's say log A times B, this can be simplified as log A plus log B.

[00:03:06]So, if we apply this to the left hand side, then we are going to have the same thing.

[00:03:13]We are going to have the same thing applied to the right as we have log six to power X being equal to log six squared plus log three.

[00:03:26]Right? Just like we have A and B, our A now is six squared and our B is three.

[00:03:32]So, from here we continue.

[00:03:36]Okay, [snorts] so from here we also know about another law of logarithm that says that the log of A to power B is the same thing as B multiplied by log A.

[00:03:54]So, if this is true, then the powers we have now are going to come down to give X log 6.

[00:04:02]Then here we have two log 6.

[00:04:06]And then we have our plus log 3.

[00:04:11]Now, the next step is that since we are trying to look for the value of X, we have to divide both sides of this by log 6 and then divide that by log 6.

[00:04:24]This can go there so that the value of X is what?

[00:04:29]2 log 6 plus log 2 log 3 rather and we're dividing by log 6.

[00:04:38]But, we can simplify further to give us X equals 2 log 6 divided by log 6 plus we got that log 3 there divided by log 6.

[00:04:55]Okay.

[00:04:59]Now, this one can go so that X will be 2 plus log 3 divided by log 6.

[00:05:11]If you apply change of base, this will change. The fraction here will go so that our X will be 2 plus log 3.

[00:05:21]And this 6 will become the base to the 3.

[00:05:25]So, this is our value of X in terms of log.

[00:05:29]Now, to verify this, we have to go back to the original equation.

[00:05:35]Okay, so this is the original equation and our X from calculation is 2 plus log 3 to base 6.

[00:05:47]This is what we have. Now, [snorts] we are going to put this value into this.

[00:05:52]But, before then, multiply your two and three both of them to the common power.

[00:06:00]Okay, so you have this.

[00:06:02]If you take a step further, you have six to power x minus 108 equals zero.

[00:06:09]So, I'm going to pick this one six to power x.

[00:06:15]This will now imply six to the power of two plus log three to base six.

[00:06:24]And from one of the laws of indices, we know that this is the same thing as six to power two multiplied by the same six to the power of log three to base six.

[00:06:39]Now, we'll use one of the laws of um of indices of logarithm rather to explain this. The law says that if you have m raised to power log n and it's to the same base m that everything there is equal to n.

[00:06:58]So, let's go back here so that we can apply what we just talked about. So, this will now mean that this and this will go and here we'll have only um three.

[00:07:11]Meaning that we have six to the power of two multiplied by three. And this means that we have 16 multiplied Okay, no. Six squared is 36, so we have 36 multiplied by three.

[00:07:27]If we multiply, what are we going to have?

[00:07:29]Six times three is 18. Take one.

[00:07:33]Three times three is nine plus one and that is 10.

[00:07:37]Okay, if you remember, what we're having is six ^ x 108 = 0.

[00:07:48]So, if this must happen, it means that this 6 ^ x is also 108.

[00:07:54]So, that 108 - 108 will be equal to 0.

[00:07:59]So, and that's the same thing we had.

[00:08:01]So, this means that we are very correct to say that x is 2 + log 3 to base 6.

[00:08:11]Thank you for watching.