Olympiad Mathematics | Germany | Can You Solve This?

追加: 2026-06-07

[00:00:02]You're ready? Let's provide a solution to this.

[00:00:07]This is an exponential equation, but then it can be confusing.

[00:00:14]- 3 to the power of x + 4 This is giving us zero. So, what do we do?

[00:00:23]We're going to apply our laws of indices.

[00:00:27]One of the laws of indices says that a to the power of m + n is the same as a to the power of m * a to the power of n.

[00:00:38]So, using this identity, I mean using this law we're going to express this as 4 to the power of x * 4 to the power of 3.

[00:00:51]Then, we do the same thing here. We have 3 to the power of x * 3 to the power of 4.

[00:00:59]This is equal to zero.

[00:01:01]Okay?

[00:01:03]So, from here now, we are going to have 4 to the power of x um multiplied by 4 to the power of 3 is 64.

[00:01:14]- Here, we have 3 to the power of x and then 3 to the power of 4 is 81.

[00:01:23]This is equal to zero.

[00:01:25]Now, if you look at this, we have x spread out. We have x here and over here.

[00:01:31]So, the first thing we do is to bring terms without x together and then bring terms with x together.

[00:01:39]So, I would divide this by 64 and divide this by 64.

[00:01:44]So, that this can take this out.

[00:01:48]And on the left-hand side, we have um On the left-hand side, we have just 4 to the power of x.

[00:01:57]Then, we have minus 3 to the power of x.

[00:02:00]Okay, we have 81 over 64.

[00:02:04]This is equal to zero.

[00:02:07]So, from here now, remember I told you we have to bring terms with x together.

[00:02:14]So, I'm going to divide all through by 3 to the power of x.

[00:02:20]Okay, so um this will cancel out.

[00:02:24]So, we now have 4 to the power of x over 3 to the power of x minus 81 over 64 to be equal to zero.

[00:02:38]Now, this is negative 81 over 64, so we can take it to the other side.

[00:02:44]So, that we can have 4 to the power of x over 3 to the power of x.

[00:02:51]Like this, will be equal to zero plus this, and that will be 81 over 64.

[00:02:59]So, what do we do next? Apply another law of indices.

[00:03:05]I told you we're going to look at laws of indices.

[00:03:09]Here, we have the same powers. So, there's a law that says if you have the same powers, you can, you know, combine the base, you know, and let them have the same power.

[00:03:21]So, we have 4 over 3.

[00:03:24]Then we have x as the power for both of them.

[00:03:28]This is equal to 81 over 64.

[00:03:32]Now, the base on the left and on the right are not the same.

[00:03:37]So, there will be need for us to take the log of both sides. So, let's do that.

[00:03:44]Okay, so like I said, we're going to take the log of both sides. So, we have log 4 over 3 to the power of x equals log 81 divided by 64.

[00:04:02]So, what do we do? Remember this law.

[00:04:06]Log a to power b is the same as log b log a.

[00:04:13]Okay, so in this case, the unknown is x.

[00:04:17]So, to find the unknown, we have to bring it down based on the law that I just explained.

[00:04:24]So, we have x and we have the log of 4 over 3 and it's equal to the log of 81 over 64.

[00:04:36]And there's something else we can do from here.

[00:04:39]You know that log a divided by b is also the same thing as log a minus log b.

[00:04:51]So, if this is true, then I'll write these two in this form.

[00:04:56]So, we have x into log 4 minus log 3 and it's equal to log 81 minus log 64.

[00:05:12]Okay, this is um done based on the law that I explained earlier.

[00:05:18]To get the value of x, divide this by log 4 minus log 3.

[00:05:24]You know, this will cancel the whole of this.

[00:05:27]But then the other side will divide by the same thing which is log 4 minus log 3.

[00:05:36]So, here we have x left and that x will be equal to log 81 minus log 64.

[00:05:49]This is over log 4 minus log 3.

[00:05:55]Now, if you decide to stop at this point, it's okay.

[00:05:59]But then, we can, you know, press a calculator and approximate the value of X.

[00:06:06]So, what do we do? Our X will be from my calculator, log 81 minus log 64 is approximately 0.10 23.

[00:06:20]This is an approximated figure.

[00:06:23]Then, log 4 minus log 3 is approximately 0.12 49.

[00:06:33]So, the approximated value of X will now be equal to Um we have 0.819.

[00:06:45]That's this divided by this.

[00:06:47]So, this is our approximated value of X.

[00:06:51]Now, let's go put it put this value back into the equation very quickly.

[00:07:00]Okay, so this is the original equation.

[00:07:04]And our value of X is approximately um 0.819.

[00:07:12]So, this means that we have 4 to the power of 0.819 plus 3 minus 3 to the power of 0.819 and plus 4.

[00:07:27]This will be equal to what? That's what we want to know.

[00:07:32]So, here we have 4 to the power of this plus this 3.

[00:07:38]3.819 Then minus here, we 3 to the power of 4.

[00:07:46]point 819. That is the addition of that.

[00:07:51]I mean the addition of the power here.

[00:07:53]So if we go on from our calculator, 4 to the power of 3.819 is approximately 199.19.

[00:08:05]199.19.

[00:08:09]Then here we have approximately 3 to the power of 4.

[00:08:13]819 is 199.

[00:08:19]18, right?

[00:08:21]So the difference here is 0.01.

[00:08:26]And ordinarily we're expecting zero.

[00:08:30]But because the value of x that we're using is an approximated value, we cannot get the exact zero.

[00:08:38]But this is still correct because the value, which is approximately 0.819, truly satisfies the equation.

[00:08:49]See you in the next video.

[00:08:52]But make sure you subscribe, you like, you comment, and you share.