Japanese | Olympiad Mathematics | Can you solve this?

Added: 2026-05-13

[00:00:00]Okay, if you're ready, I am ready. Let's solve this one quick one.

[00:00:08]This is five to power x - 300 = 0.

[00:00:16]Now, what do you do?

[00:00:18]What will be the first step you're going to take to remove this 300? So, we have 5 to the^ x - 300.

[00:00:28]It's already negative, right? So to remove 300 now we are going to add 300 again and we'll add the same to the right. So we have 300 there. This is called the additive inverse of minus 300. So this and this will go and now we have our 5 to the^ x to be equal to 3000.

[00:00:53]Now how do we work this? Can we express 300 in the base of five completely?

[00:01:00]The answer is no. So let's bring out the five that we can find from 300.

[00:01:07]5 to the^ x is equal to now 5 is a factor of 300. So I'm going to bring out five then multiply by 5 into 30 is going to be 6.

[00:01:22]Right? And then 5 into 0 is going to be 0. So 50 5 * 60 will give 300.

[00:01:31]We can bring out another five from 60.

[00:01:35]Yes. So we're going to have 5 to the^ of x to be equal to 5 * 60 is 5 * 12.

[00:01:47]Yes. So 5 * 12 will give 60. So as a matter of fact our 5 to^x will now be 5^ 2 * 12.

[00:01:59]So we are not able to have the same base completely on both sides of the equation.

[00:02:06]And this is the point where we'll take the log of both sides. So if you're ready let's go.

[00:02:15]This is log 5 ^ x to be equal to log 5^ 2 * 12.

[00:02:26]Yes. So this is what it is. And from here we can apply a law.

[00:02:32]Okay. Because this is 5² * 2 * 12. And if we have log a b this can be expressed as log a then plus log b separately log a + log b separately. So we now have our log 5 to the^ of x and what we have here will be expressed in this form which is log 5^ 2 then + log 12 Yes. So we have log 12 over over there.

[00:03:12]Now we are having something like we're having powers too. Power X power two.

[00:03:17]And there's a law of logarithm that is related to the power. Okay. If you have the log of um p okay let's say m to the^ n this is n log m. Take note of that.

[00:03:33]So what happens is that the n comes down to multiply the log. Same thing happens over there as we have x log 5.

[00:03:45]Then here there's power of two. So the two will come down. We have 2 log 5.

[00:03:50]Then we have our plus log 12.

[00:03:55]So from this point we go on.

[00:04:00][snorts] Okay. So what we are doing now is to remove log five so that x will become the subject of the okay will become the subject. So divide both sides by log five the same log five.

[00:04:19]log five will cancel itself from the left and we have x to be 2 + okay by the way that is 2 log 2 log 5 + log 12 / log um log 5. And do not forget that if you have something like this m + n over let's say a this can be written as m / a + n / a. So I'm going to write this in this form.

[00:05:02]But before then let me remove this so that we can have enough space. Now we have our x as 2 log 5 / log 5 then plus log 12 divided by log 5 the same log 5 so that log 5 can cancel itself and two remains there. So x will now be 2 + log 12 / log 5. So this is the value of x. But we can do something here.

[00:05:48]Let's remove this fraction from here. To remove that fraction, we will now have x [snorts] to be equal to 2 + log 12.

[00:06:00]This five becomes the base. So this is now our value of x. But then to clear the doubt we are going to verify. So let's put this value of x into the original equation.

[00:06:17]Okay. So this is the original equation and our value of x from the calculation is 2 + log 12 to base 5. Right? So what we'll do now is to put in the value to put in the value of um x into this.

[00:06:38]So we now have 5 to the power of 2 + log 12 to base 5. Then we have - 300. So let's see if this is going to give us zero. If it doesn't give zero then we are not correct.

[00:06:57]From here apply one of the laws of indices.

[00:07:01]So we have 5 to the^ of 2 multiply by the same five to the power of log 12 to base 5.

[00:07:13]Okay? Because if you since this is multiplication pick one of the bases which will give you this five then add the two powers - 300.

[00:07:24]Then we need to understand another law.

[00:07:27]If you have m to the power of log n but it is to the same base of m same base right. So the whole of this is equal to n.

[00:07:41]And now look at this and what we have over here.

[00:07:47]Okay. Compare this to this. So if the whole of this is n then the whole of this is going to be 12. Take note of that. So let me remove this.

[00:07:59]Let me remove this one here.

[00:08:02]So we are going to have 5^ squ is 25 multiply by we say this and this are going. So we have only 12. Then we have - 300.

[00:08:15]So we multiply 25 by 12 and that will give us 300. 300 - 300 will give us zero and that is what we had on the other side of the equation. Remember the equation is 5 to the^ x - 300 = what? Z. So we have confirmed that x to be = 2 + log 12 to base 5 truly satisfies the given equation. Thank you for watching.