Olympiad Mathematics | This is beautifully solved | Japanese can you?

追加: 2026-05-05

[00:00:01]Hi.

[00:00:03]Let me show you how to solve this problem here.

[00:00:09]Solution.

[00:00:12]We have 4 to the power of x over 8 equals 6.

[00:00:20]Okay. What should be the first step?

[00:00:23]We're going to take We're going to cross multiply, right?

[00:00:29]So that we can have 4 to the power of x to be equal to 8 multiplied by 6.

[00:00:41]Okay, 8 multiplied by 6.

[00:00:45]And if we multiply that, we're going to have 48. 8 * 6 is 48.

[00:00:55]And now you look at what you have on the right.

[00:00:59]When you If you write this in index form, it's not going to give you base 4 completely.

[00:01:06]So there'll be need for you to take a step.

[00:01:10]So what we'll do now is to bring out 4 from 48. 4 into 4 that will give 1. 4 into 8 is 2.

[00:01:21]This means that 4 * 12 is 48.

[00:01:25]We can have another 4 from 12.

[00:01:29]So have 4 to the power of x to be equal to 4 * 4 * 3.

[00:01:39]4 * 3 is 12. So from there now take a step and get 4 to the power of x to be equal to 4 squared * 3.

[00:01:53]Okay, because 4 * 4 is 4 squared.

[00:01:57]So this is not completely in the base of 4.

[00:02:02]So we'll take the log of both sides and we'll have log 4 to the power of x to be equal to the log of 4 to the power of 2 * 3.

[00:02:19]Right?

[00:02:20]So if you don't know about your logarithm, just stay here and know a lot about logarithm.

[00:02:25]Now do you know that if you have log CD this is the same as log C * D, right?

[00:02:34]Now from one of the laws of logarithm, we can separate this to get log C plus log D.

[00:02:46]Okay?

[00:02:47]So if we do that, that means >> [sighs] >> that means that what we have here now is going to be written in the form of this.

[00:02:58]So we have log 4 to the power x to be equal to log 4 squared plus log what?

[00:03:10]log [clears throat] 3.

[00:03:14]Now we take a step further, talk about the powers.

[00:03:19]log C to the power D is the same thing as D log C from one of the laws of indices.

[00:03:29]So this means that the power will come down to multiply the log.

[00:03:34]And here we have two powers. We have x and 2.

[00:03:38]Both of them are going to go behind.

[00:03:43]Okay, so if they go behind, we have x log 4 equals 2 log 4 plus log 3.

[00:04:00]Like this.

[00:04:03]And uh remember that we're trying to get the value of x. So we can divide both sides of the equation by log 4.

[00:04:12]Then divide the whole of this by log 4.

[00:04:16]So this one will go with this.

[00:04:19]And only x remains on the left-hand side.

[00:04:24]Okay, so this is what we have.

[00:04:27]And uh we take a step to get x equals 2 log 4 divided by log 4 plus log 3 divided by the same log 4.

[00:04:46]So if we proceed this one will just take this out.

[00:04:51]So we have x and this 2 remaining over there plus this. This is log 3 divided by log 4.

[00:05:04]Okay, so I believe you know about change of base, right?

[00:05:08]That if you have log M divided by log N you can write this as log M to base N.

[00:05:21]Okay, that's one of the laws of logarithm.

[00:05:24]So we'll apply the same law to what we had earlier.

[00:05:31]And we're going to have um okay.

[00:05:34]We're going to have x We're going to have x to be equal to 2 plus the same thing, 3 to base 4. So we have log 3 to base 4.

[00:05:52]Okay, so this is what we have as our value of x.

[00:05:59]But then I would like us to go back to the original equation and put this value.

[00:06:05]Yes, so that we can be very sure.

[00:06:10]Okay, so this is the equation.

[00:06:13]And our value of x is 2 plus log 3 to base 4. So let's put it into this problem now and see.

[00:06:25]You know what? Let me just pick Let's pick 4 to the power of x and deal with it. Then whatever we have we divide by um 8.

[00:06:34]So 4 to the power of x becomes 4 to the power of 2 plus log 3 to base 4.

[00:06:48]Okay, now this is 4 to the power of 2 multiplied by 4 to the power of log 3 to base 4.

[00:07:02]Because from one of the law uh one of the laws of indices 4 and 4 you can pick one of the bases and then you add the powers because this is multiplication.

[00:07:14]And um another law again is that if you have if you have um B to the power of log A to base B this is equal to A.

[00:07:29]So we we will now apply this to this particular part. So the whole of this is going to be 3.

[00:07:37]And um this means that we have it.

[00:07:40]We're going to have 4 squared * 3.

[00:07:43]And that is 16 * 3.

[00:07:46]What is 16 * 3?

[00:07:49]That is 48, right?

[00:07:51]Remember the given equation is 4 to the power of x over 8 equals 6.

[00:07:58]And now our 4 to the power of x is what gave us 48. So we're having 48 48 over 8. And that is the same thing as 6, right? So that means that we are very correct to say that the value of x is 2 plus log 3 to base 4.