USA | Math Olympiad 4^(x-2)=3 | Solve And Verify x.

Added: 2026-05-24

[00:00:00]Hello and welcome to another math class.

[00:00:04]Yeah, this question here is actually very simple but the verification is what is interesting and so watch from the beginning to the end because at the end of the day we're going to verify the answer we're going to get from Vienna.

[00:00:17]Okay, we are not going to give our answer in uh numeral. It's going to occur in um log form. But that love form, we're going to use it to verify if the answer actually satisfies the original equation. And it's amazing. And so stay from the beginning to the end of this math class. Welcome on board. So let's take our solution. We are looking for x here. Okay.

[00:00:39]So here we have the question is 4 ^ x - 2 = 3.

[00:00:46]Wow. So how do we solve this? Now look at this. Here there is something I want to do here. I want to rewrite this part of this equation. Applying the law in English says that your um your let's take our a to the power of m - n is equal to your a to the power of m* the a to the^ minus n. Right? And again we're still going to use another log which says that your a to the power of minus m is equal to 1 all over a ^ m. And so with that we just have to rewrite this part. So this is going to give us 4 to the^ of our x then times our 4 to the^ of -2 = pos3.

[00:01:33]Simple right? Good. Now from here we have to rewrite this expression we have in here. And so this going to give us applying this law here. So this is going to give us our 4 ^ x * our 1 all 4 ^ 2 = 3. 4 ^ 2 is 16. But we don't want to simplify that down. Let's keep it here.

[00:01:57]Okay. So the next thing we do want to multiply through by 4 to the^ of 2 to eliminate this 4 to the^ of two. And so this going to give us 4 ^ x * 1 all 4 to the^ of 2 * our 4 the^ 2 = 3 * our 4 to the^ of two. Easy. So this will go with this. We're now left with 4 the^ x is equal to 3 * 4 to the^ of two. Good. The math is now getting interesting from this part here. Now look at what I want to do here. Here I don't want to find the value of this then multiply this have a figure then do whatever whatever but yeah I want to log both side of the equation the question now is to what base am I going to take my log here now yeah I want to take my log to base four okay instead of b 10 okay so yeah going to have here to be our log 4 to the^ of x is 4 then equal to our log 3 * 4 to the^ of 2. Okay. Or in base 4.

[00:03:07]Let me put this in bracket. Wow. This is interesting. Right. Good. Then from here I want to use I want to apply the law of logarithm to this. The two laws of logarithm we are going to use is the multiplication law and the power law of logarithm. The multiplication of logarithm says that when you have your log m yeah times n this is both of your log m plus your log your n. We're going to make use of that for this expression here. Therefore the power law we're going to make use of your log your m to the power of your a. said move this a this way to give us here a * log your m.

[00:03:53]Wow. So first of all let's open up this.

[00:03:55]So this going to give us log log our four b to the power of x b 4 equal to the log of our three b 4 plus the log of our 4 to the^ of two means four. Easy.

[00:04:14]So let's rule out this here and see what this gives us a third. So we moving the whole of this this way and we are moving the whole of this this way. So if we do that we're going to have our x * log 4 b 4 = to we're going to have log 3. So we have log 3 b 4 plus our 2 * log 4 b 4.

[00:04:40]Easy.

[00:04:41]Okay. Now we're going to apply another law again which says that if you have your um if you have your okay let me put it here. If you have your log a b a this is equal to your one provided a is not equal to zero.

[00:05:00]And so here what we have here is four not zero and also not imaginary number.

[00:05:06]And so what happen? This is going to give us our x = log 3 base 4 + our 2.

[00:05:15]Easy. So rewriting this is going to have here 2 = 2 + our log 3 4.

[00:05:24]But we can still simplify this down.

[00:05:27]Listen and watch this carefully. There is a law that is not commonly used in logarithm at some level. Okay, that is what I want to bring in here. Now look at this four here. I can write this four as 2 to the^ of two. And so this going to give us our x = 2 + log 3 is 2 to the^ of 2. Wow. I'm going to apply a law here. The law which says that if you have your log uh let's take your log a b's um let's take b to the power of c.

[00:06:01]So this is equal to 1 all over c * log your a base b. Wow. So I'm going to apply that here. And so this is going to give us our x = 2 + our 1 all / 2 * our log 3 b 2.

[00:06:19]So this is the value of x. Now remember I can rewrite this. I can convert this to base 10 applying the change of base law. Right? goes this five. To do that, this is going to give us x is equal to the log which says that if you have log your a base b, change it to b c. This is going to give us log your a b c all over log your b c.

[00:06:48]You see? So I can change this base to base 10. But again, this is our answer, right? Whatever I'm going to do here is still our answer because we cannot get this in your base calculator. And so if we want to express this in base 10, then this is going to give us our two plus our 1 all 2 times.

[00:07:12]Yeah, we're going to have log 3 b 10 all over log 2 b 10. These are simple log. You can impute in your calculator to get your values. Right? Good. Now from here again this is also your answer. But we want to do a verification want to do a check on this um solution either this or this to see if actually satisfy this equation.

[00:07:42]So let's take our check. Let's roll off and take our check here. Check. So the question is 4 to the^ of x - 2 = r 3. So wherever we see x we're going to put in the whole of this value with this actually give us three. So let's see there. So this going to give us our 4 to the power of um where's the value? We have 2 + 1 all over 2 * log our 3 base 10. Everything equal to three.

[00:08:17]Okay.

[00:08:19]All right. Um, yeah. Uh, okay. Minus two. I must carry. Yeah. Because this is our answer. Then we have minus2 here.

[00:08:28]So, -2 close bracket equal to 3. Okay.

[00:08:33]So, with this what happen? We having a + two and here we have minus2. So, this will go with this. So we're now left with your 4 to the^ 1 all / 2 * our log 3. What am I right here? Is it base 10?

[00:08:49]Oh, pardon me. It is base two, please.

[00:08:52]Base two. So we have here base two.

[00:08:54]Okay. Equal to our three. Okay. So what we do next here is this. We can rewrite this our four. Four could be written as 2 to the^ of two. So we have 2 to the^ of two. the or in bracket 1 all over two * log our 3 base 2 close bracket everything equal to 3. We can use this to open this. Applying the law we say that if you have your a to the power of m n this is same thing as a to the power of m or in bracket raised to n which is also equal to your a to the power of n or in bracket raised to n good so from there you discover that this will go with this. So when I have 2 to the power of log 3 base 2 everything equal to three. Now look at what we have here. There is another law that says that if you have your a to the power of your log let's take b's a say this is equal to your b right and so what happened this and this and so we're going to have three is equal to 3.

[00:10:04]So this shows that the answer we solve for the root we solve for which is this or this actually satisfy the original equation. So this is how you solve and verify simple exponential equations of this kind. Thank you for watching. Drop a question in the comment section. We meet you over there in the comment section. Bye. We love you. Keep watching online TV. Bye for now.