Math Olympiad|Exponential math problem.

Hi, welcome to my channel. Let's find the value of x. I would like you to watch to the end because this is not an easy task. Okay, now let's do it together. The first thing we need to do is to divide through by 9 to the power of x. in order to simplify this equation Okay, so divided by 9 to the power of x, divided by 9 to the power of x divided by 9 to the power of x Okay, so when we do that, we are going to have 1 plus 15 to the power of x. is equal to 25 raised to the power of x, sorry, divided by 9 to the power of x This is divided by 9 to the power of x. Now, recall that a to the power of x, divided by b to the power of x can be written as a divided by b everything raised to the power of x So we are going to apply that here. So when we do that, we will have 1 plus 15 over 9 to the power of x is equal to 25 over 9 to the power of x. Alright, now we have 1 plus, This and this—they have this common factor, right? Which is 3. So when 3 divides 15, we get 5. And when 3 divides 9, we get 3. So we have this to the power of x is equal to, now here we have, 25 is the same as 5 squared, while 9 is the same as 3 squared, raised to the power of x Please subscribe to the channel if this is the first time you're encountering this channel. and also turn on your notification bell so that you will be notified whenever I upload a new video. Thank you so much. So at this point, we are going to bring this 2 out. Let's bring it out. So we have 1 plus 5 over 3 to the power of x, right? Okay, is equal to 5 over 3, squared then raised to the power of x. Now, look at what we have. Record that a to the power of m raised to the power of n can also be written as a to the power of n raised to the power of m, okay? That's why I'm going to write that this is 1 plus 5 over 3 to the power of x. is equal to 5 over 3 to the power of x squared, okay? Now, let 5 over 3 to the power of x be equal to y. Yes, since we have the same thing here and here. This will allow us to form a quadratic equation. But trust me, after solving the quadratic equation, the main thing, the most interesting aspect of this particular video It is where we have to bring this in order to find the value of x. Yes, okay. Now, since we let this be equal to y, Now let's replace or substitute. So we are going to have 1 plus y, right, is equal to y squared. We have simplified it. Now we have a quadratic equation. What should we do? Let's organize it properly. So to arrange it properly, your y(graduated) squared comes first. So this will transfer to become minus y. Why this will transfer to become minus 1 is equal to 0. Hope you have subscribed. Have you? Alright, this is what we have here. So we are going to use the quadratic formula to solve for the value of our y. Then substitute it back into this to get the value of our x, okay? So now we have that y squared minus y minus 1 equals 0. Here, our a is the coefficient of y squared, which is 1. Our b is the coefficient of y, which is minus 1. Our c is the constant which is minus 1. Now let's write the quadratic equation formula which states that this y will be equal to minus b plus or minus square root of b squared minus 4ac Everything divided by 2a, okay? So we have that y is equal to minus b. That is minus minus 1, which is 1. plus or minus the square root of b squared That is minus 1 squared, which is 1. -4 × 1 × -1 Which will give us minus 4, right? Now minus minus will give us plus. So this becomes plus 4 divided by 2a. That is 2 times 1, which is 2. That means our y is equal to... Okay, this place is small. Oh, it's okay. y is equal to 1 plus or minus the square root of 5, divided by 2. Now this is where I want you to start paying attention. because this is where it actually gets really interesting, okay? Remember that I upload two videos every day. So whatever you are doing, whenever you have the opportunity or the chance, Please click on the channel. Go to my page and see what I uploaded, okay? Thank you. Now this is what we have. From here we have two values of y. The first value is that y is equal to 1 plus the square root of 5 over 2. Now the second value is that y is equal to 1 minus the square root of 5 over 2. Okay, so but recall, We have to recall what? 5 over 3 to the power of x. 5 over 3 to the power of x is equal to y. That means we are going to replace this with this. to solve for x because that is what we are asked to find. Let me clean this side of the board so I can do that, okay? So if I were to do that, I will have... Okay, this is our y. So this is what we have here. The one we want to solve. That y₁ is 1 plus the square root of 5, divided by 2. Now since y is equal to 5 over 3 to the power of x, So we are going to write 5 over 3 to the power of x. is equal to 1 plus the square root of 5 divided by 2. Now when you look at this, Notice that the only savior we have is to introduce our log reading, right? So let's go ahead and do that. We are going to have log base 5 over 3 raised to the power of x. is equal to log of 1 plus the square root of 5, divided by 2. Alright, I can clean this now. I don't think I need them anymore except for the second value of my y, okay? So I will leave that. Now this is what we have. Now I want us to recall our logarithmic thing. Our logarithmic law, the laws of logarithmic reading. That means that the logarithm of a raised to the power of b is equal to b multiplied by the logarithm of a, okay? So we are going to do that here. So we have x multiplied by log 5 over 3 is equal to log of 1 plus the square root of 5, divided by 2, alright? Now, we are looking for the value of x. So let's divide both sides by log 5 over 3. So we are going to have that x, x log(5/3) divided by log(5/3) is equal to what? log(1 + √5 / 2) divided by log(5 / 3) Now log 5 over 3 will divide itself. So we have that x is equal to. Now I want us to recall that log a divided by log b. Once they're in the same base, right? Can be written as, I just want to leave my Y too. Okay, can be written as log base b of a, okay? So this becomes the base. Therefore, I know it's the opposite of this one. Okay, so therefore we have that x is equal to what? Log of 1 plus the square root of 5, divided by 2, okay? Five over three. Now this is the first value of our x. Yes, so this becomes the first value of our x. Now for the second value, We're going to use the same method, alright? Now recall that our y in that second value, Our y is equal to 1 minus the square root of 5, divided by 2. That is our y. But note that when we solve this, We're going to get a negative value, okay? When we solve this, we are going to get a negative value. So if we say that our x₂ is log of (1 minus root 5) over 2 Five over three in base 5 doesn't exist. Because they cannot find the logarithm of a negative number. That brings us to the conclusion. that this is the only value of our x. Thank you so much for watching, and have a lovely day.