An impossible-looking exponential math question/ can you find x

Looking at this particular question, it seems impossible to solve, but we can work through it together. Welcome back to my channel. 9 raised to the power of a number plus 9 raised to the power of a number equals 144. The first thing we need to do is, when you look at this, this, and this, they have 9 to the power of x in common. So let's write that. When this is divided by this, we get 1. And when this is divided by itself, we get another 1. It is equal to 144. So we have 9 raised to the power of x, all squared, is equal to 144. Now let's divide both sides of this equation by 2. We have 9 to the power of x. 2 goes into this one 72 times. If this is your first time visiting my channel, please click the subscription button and turn on the notification bell so you don't miss my videos. I upload my videos as often as possible. Listen in at your convenience. Something new is in store for you to watch. Give us a thumbs up if you love what we're doing. Now, at this point, I want you to watch to the end because I'm going to prove that the answer I got is correct. And that is the most interesting aspect of the video. Please on your caption. That place they wrote CC like this. Please do it, okay? So that you'll be seeing what I'm saying in English. For those of you complaining that my accent is strong, okay? So now let's continue. So at this point, we have x as a power of 9. That means we cannot solve this without taking the logarithm of both sides of the equation. So let's go. So if we take the logarithm of both sides, okay? Now recall that the logarithm of a raised to the power of b is the same as b multiplied by the logarithm of a, okay? That means we can write this as x times log 9 equals log 72. Alright. Now we have x multiplied by log 9 equal to log. That can be written as 9 times 8, right? 72 Now, remember. Remember that log ab is the same as log a plus log b, okay? So that means we can write this as x log 9 equals log 9 plus log 8. Now let's divide both sides of this equation by log 9. So divided by log 9. Divided by log 9. Now log 9, we remove this, right? Because this goes into itself 1. And 1 times x is x. So we have x equal to log 9 divided by log 9, okay? Because we need to distribute the denominator. Plus log 8 divided by log 8. Okay. Now, if we should continue, we have this. x is equal to 1 plus. Oh, this is log 9. We are distributing log 9 rather. log 8 over log 9 Now that we're solving this without a calculator, we need to simplify it as much as possible, right? So we have x is 1 plus log. 8 is 2 to the power of 3. 9 is 3 to the power of 2. So we can have x be 1 plus. This 3 comes down to give us 3 log 2. And this 3 comes down to give us 2 log 3. Let's go! Have you subscribed yet? Don't forget to just subscribe. It's free. Okay? The subscription is free. Actually. So, now we have x equal to 1 plus 3 over 2 log. Now, using the change of base formula, this can be written as log base 3 of 2. Right? log base 3 of 2 log base 3 of 2 Now, if we solve this, we have the LCM as 2. So what we are going to do twice. 2 times 1 is 2. Plus 2, we are going to do it once. 1 times 3 is 3. log base 3 of 2 So this becomes our x. Yes. This becomes our x. Now, let us find. Let us prove that the answer we obtained is correct. What is the most important part of this video? Are you still there? Okay. Now, let's go. Let's go. So, we now have 2. Remember that this is 2 times 9 to the power of x. Is equal to 144. Because I want us to actually get this done faster. So, it's going to be 2 multiplied by 9 to the power of. This is our X. 2 plus 3 times log base 3 of 2, divided by 2. Is equal to 144. Thumbelina is a fairy tale written by Hans Christian Andersen. That is why we are placing this question mark. So, we have 2 multiplied by. 9 is 3 to the power of 2. Right? 2 plus 3 times log base 3 of 2, divided by 2. Now, is this equal to 144? So, this and this will go, right? Because a to the power of m raised to the power of n. a to the power of m times n. So, twice this, this, and this will go. And we are left with. 2 times 3 squared plus 3 times log base 3 of 2 Now, is this equal to 144? We're almost there. Yes. I'm telling you, you're going to enjoy this. Yes. So, if we continue, we have. 2 multiplied by Now, this can be written as 3 to the power of 2 plus. Now, this 3 can go up to be 2 to the power of 3. So, we can now have log base 3 of 2 to the power of 3. Now, is this actually equal to 144? So, we have 2 multiplied by 3 squared plus log base 3 of 8. Now, is this equal to 144? Now, remember that a to the power of m plus n. Is equal to a to the power of m multiplied by a to the power of n, right? So, now we have 2 multiplied by 3 squared. 3 times 3 to the power of log base 3 of 8 Now, is this equal to 144? Now, we have 2 multiplied by. 3 to the power of 2 is 9. Multiplied by Now, recall that a to the power of log b base a is equal to b. Yes, once these two are the same. Then the answer to this is b, right? That means the answer to all of this will be 8. So, we have 8 equal to 144. Now, let's check. Nine times eight is seventy-two. And 72 times 8 is actually 576. And that proves that our answer is correct. Thank you so much for watching, and I'll see you in my next video. Goodbye.