Olympiad Mathematics | Japanese | Can You Solve This One?

追加: 2026-05-22

[00:00:01]If you're ready, let's provide a solution to this problem here.

[00:00:06]Solution.

[00:00:10]We have 5 to the power of x plus 5 to the power of x equals 30 over 7.

[00:00:19]Okay, um what do we do from here?

[00:00:22]We're going to add the left-hand side, which is 5 to power x in two places.

[00:00:30]And it's equal to 30 divided by 7.

[00:00:35]Now, the next thing you should do is to make 5 to power x to be alone.

[00:00:41]Meaning that we need to remove this two from here.

[00:00:44]And to remove that two, you divide by two.

[00:00:48]And then you divide the right-hand side by two.

[00:00:51]This will go.

[00:00:53]So that we can have 5 to the power of x to be equal to 30 over 7 multiplied by 1 over 2.

[00:01:03]Because if you multiply any number by 1 over 2, you're equally dividing the number by two.

[00:01:10]So two can now go into that to give us 15.

[00:01:14]And we'll have 5 to the power of x to be equal to 15 over 7.

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[00:01:35]Okay, so at this point now we are looking for a way to actually make um the base to be equal, but the base cannot be equal, right? Since we cannot equate the base, we will have to take the log.

[00:01:55]We'll take the log of both sides. So, we have log 5 to the power X to be equal to log 15 over 7.

[00:02:09]Okay? Then, let's apply this law that log A over B is the same thing as log A minus log B.

[00:02:21]This is one of the laws of logarithm.

[00:02:23]So, if we apply this law here to the right, we'll have log 5 to the power X to be equal to log 15 log 15 plus log 7.

[00:02:41]But, then there's something we can do.

[00:02:43]There's something that we can do because this 15 here is the same as 5 * 3. Since we have 5 here, let's break that 15.

[00:02:54]Okay.

[00:02:56]So, oh, I think I made a mistake. It's 15 over 7, right? So, that means this is going to be negative.

[00:03:02]So, this should be negative.

[00:03:04]Okay? And then, here's what we can do.

[00:03:07]We're going to break the 15. So, we have our log 5 to the power X to be equal to log 5 * 3. Then, we have our minus log 7.

[00:03:20]Then, from one of the laws of logarithm, log MN is equal to log M plus log N.

[00:03:30]So, we'll apply this to this place and we'll have log 5 to the power X to be equal to log 5 plus log 3 minus log 7.

[00:03:48]Okay? So, this is what we have.

[00:03:51]And we will not stop here, right?

[00:03:53]Because we are looking for the value of X. And there is a law that says we can always drop down the power. So, the power come down, we have X log 5 which is equal to log 5 plus log 3 minus log 7.

[00:04:12]>> [snorts] >> Okay, at this point, we are going to divide all through by log 5.

[00:04:21]So, that this can take this out and X will be free as it is equal to log 5 plus log 3 minus log 7 divided by log 5.

[00:04:37]Now, I believe you know about this, right?

[00:04:40]If you have A plus B plus C all over D.

[00:04:46]What does this mean?

[00:04:47]It means that you're having A over D plus B over D plus C over D, right? So, it doesn't matter if this is negative or positive, right?

[00:05:01]So, I'll apply the same thing to the right hand side.

[00:05:09]So, that we can have our X to be log 5 plus Okay.

[00:05:16]log 5 over log 5 plus log 3 over log 5 then minus log 7 over log 5.

[00:05:31]Okay, so log 5 in the three places, right?

[00:05:36]Then, let's apply change of base. If you apply change of base, 5 here becomes the base to the three of them. So, our X will now be log 5 to this base, which is 5. Then we have plus log 3 to base 5 minus log 7 to the same base of 5.

[00:05:58]This is one of the laws of logarithm.

[00:06:00]And then the other law of logarithm also says that log 5 to base 5 should be 1.

[00:06:08]Log of the same number to the same base is 1. So our X is going to be 1 plus we have log 3 to base 5 minus log 7 to base 5.

[00:06:22]Now, let's stop here and call this for value of X. But if you want to have decimal figure, you can, you know, press your calculator and get your answer in decimal form.

[00:06:34]Now, let's verify to be sure we are correct very quickly.

[00:06:40]Okay, so this is the equation and the value of X we have is 1 plus log 3 to base 5 minus log 7 to base what?

[00:06:57]To base 5. Okay, let's go over log 7 to the base of 5.

[00:07:03]So I'm thinking what to do. Now look at the left-hand side here. Let's write two multiply by 5 to the power of X.

[00:07:14]Okay.

[00:07:15]So and that will be equal to 30 over 7.

[00:07:20]So that writing 5 to power X in two places, let's do it this way.

[00:07:24]Now in place of that X, I'm going to put in the value so that we have two into 5 to power 1 plus log 3 to base 5 minus log 7 to base 5.

[00:07:45]Now, let's see what this will give us.

[00:07:48]And we'll apply this law here that says um a to the power of um b plus c even if we have minus c, right? This can be Okay, minus d.

[00:08:01]This will be expressed as a to power b times a to power c times a to power of minus d.

[00:08:11]It's the same thing. So, I will do the same thing now.

[00:08:16]Okay, so if I do the same, I will have to multiply by 5 to power 1, multiply by 5 to the power of log 3 to base 5, then multiply by 5 to the power of negative log 7 to base 5.

[00:08:34]Now, take note of this. Take note of this law.

[00:08:38]The law says that the law says that a to the power of log b but to same to the same base of a.

[00:08:47]This and this are going to go and you'll be left with only b.

[00:08:51]Okay?

[00:08:52]So, if I apply this Okay, before I apply that, let's do this. We're going to have to multiply by 5 to power a is 5.

[00:09:02]Then multiply by 5 to the power of log 3 to base 5.

[00:09:08]Then multiply by 5.

[00:09:10]Okay, look at this. We need to remove this negative. So, we have 1 over everything, which is 5 to the power of log 7 to base 5.

[00:09:24]Interesting, right?

[00:09:26]So, that from here, we'll now apply the law that I explained earlier. This is going to go with this and we have to we have to to multiply five to multiply three since this and this have gone. Then to multiply one over this and this will also go. So, it's one over seven.

[00:09:49]So, we do this. Now, let's go. We have two times two times five times three. Two times five is 10. 10 times three is 30. 30 times one is 30. So, we have 30 over this seven.

[00:10:07]Okay. No wonder the equation we have solved is five to power X plus five to power X equals 30 over seven.

[00:10:17]The same 30 that we we just got.

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