Olympiad Mathematics | Germany | Can You Solve This One?

Added: 2026-06-09

[00:00:01]If you're ready, let's solve this one.

[00:00:07]6 ^ x + 6 ^ x = 300.

[00:00:15]Okay, so how do we solve this?

[00:00:19]You do not introduce log. Okay, at this point.

[00:00:24]What should you do?

[00:00:25]Um here we have 6 ^ x as a common factor.

[00:00:31]Then 1 + 1 in here.

[00:00:34]6 ^ x * 1 6 ^ x.

[00:00:37]6 ^ x * this will give that.

[00:00:40]So all of this is equal to 300.

[00:00:46]Now 6 ^ x into 2 is equal to 300.

[00:00:53]So there'll be need for us to divide both sides by 2.

[00:00:58]Right? So that this can go. Then here we divide by 2.

[00:01:04]Now on the left-hand side we have 6 ^ x.

[00:01:08]On the right-hand side we have 150.

[00:01:12]You know, that is the half of 300.

[00:01:17]Okay, so at this point we look at the right-hand side.

[00:01:22]150 cannot be written in the base of 6.

[00:01:25]Yes.

[00:01:27]But then let's see. Can 6 go into 150?

[00:01:31]The answer is yes.

[00:01:33]>> [snorts] >> 6 into 15, what will that give us? It will give for 2.

[00:01:38]Right? Remember 2 * 3 um 2 * 6 is 12.

[00:01:43]That means there's a remainder of 3.

[00:01:45]Making the remaining zero to be 30.

[00:01:48]30 / 6 will give us 5. So this is 150.

[00:01:54]Now the only thing we can do is to take the log of both sides.

[00:01:59]And we have log 6 to the power of x equals the log of 6 multiplied by 25.

[00:02:10]And then, from this very law log AB is the same as the log of A plus the log of B.

[00:02:21]And um AB here is the same as A * B, just like what we have over there.

[00:02:27]So, we have our log 6 to the power x to be equal to log 6 plus log 25.

[00:02:38]This is interesting, right?

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[00:02:53]Now, the next thing we're going to do is that we are going to apply a law that has to do with the power.

[00:03:03]You know, there's a law that says the power here can always go behind.

[00:03:06]So, we have x log 6 and is equal to log 6 um plus log 25.

[00:03:19]And then, at this point, we are trying to get the value of x. So, the only thing stopping us from doing that is the log 6.

[00:03:27]So, we have to introduce log 6 to the left and to the right, we introduce the same log 6.

[00:03:35]And now, x on the left is equal to what we have here.

[00:03:41]Okay, so from here, we will simplify to get x equals log 6 divided by log 6 then plus log 25 divided by the same log six.

[00:03:58]Now we have X equals log six over log six is one.

[00:04:03]Then we have log 25 divided by log six.

[00:04:10]And if you know about change of base, what would you do?

[00:04:14]I hope you know that if you have log B over log A This is the same as log B to base A.

[00:04:25]Right? So if this is true then we can write this as X equals one plus log 25 to base six.

[00:04:37]This six becomes the base to the 25.

[00:04:43]So now we have our value of X.

[00:04:48]But we'll verify to be sure that our works we've done is correct.

[00:04:53]If you remember, the equation is six to the power X plus six to the power X equals 300.

[00:05:02]And again, you should know that we simplify the left hand side as we get two of six to the power of X.

[00:05:12]But in our workings, I think we got six to the power X into two.

[00:05:18]But this is still the same thing and it should be equal to 300.

[00:05:22]Now if we put in the value of X, are we going to have the same 300 on the right?

[00:05:29]That's what we want to find out.

[00:05:31]This is two into bracket six to the power of X which is one plus we have log 25 to the base of six.

[00:05:42]And we want to see whether this will satisfy the equation perfectly.

[00:05:50]All right, so we still have two on the outside.

[00:05:55]And here we apply one of the laws of logarithm.

[00:05:59]This is 6 to the power 1 multiplied by the same 6 to the power of log 25 to base 6.

[00:06:08]Right?

[00:06:09]Now, [snorts] this is two multiplied by 6 to the power 1 is 6.

[00:06:15]Then you look at this.

[00:06:17]This is 6 to the power of log 25 to base 6.

[00:06:21]Now, the base here is 6 and the base to the log is also 6. So, both of them will undo each other. So, we have 6 to the power 1 * 25.

[00:06:32]And that is the same as 6 multiplied by what? 25.

[00:06:37]Okay, so let's simplify this.

[00:06:40]Okay, so from here we have two multiplied by 6 multiplied by 250.

[00:06:48]I mean, 6 multiplied by 25 will give us 100 and 50.

[00:06:54]150 multiplied by two will give us 300.

[00:07:01]Right?

[00:07:02]And if you remember, the equation that we've solved is 6 to the power of x plus 6 to the power of x equals 300.

[00:07:12]So, this means that the value of x that we got is um very satisfying. Now, what is even the value of x that we got?

[00:07:22]Okay, so our value of x is 1 plus log 25 to base 6.

[00:07:28]This is in log form.

[00:07:30]If you want to get your value in um in decimal form, you have to press calculator log 25 to base 6. Then whatever you have, you add one.

[00:07:41]So, you now have your answer in decimal form, which you will approximate yourself. Think we should do this, right?

[00:07:51]Okay, so from calculator, our X is approximately 2.796 79 6. Okay, so this is an approximated figure, and this is um our answer in log form.

[00:08:09]Thank you for watching.

[00:08:11]And um if you have not subscribed to this channel, consider subscribing so that you will be notified whenever we have new videos.

[00:08:22]By the way, this is nine.

[00:08:24]2.796.

[00:08:26]Thank you.