Nice exponent simplification| math Olympiad| solve for x.

Welcome, my lovely viewers. Yes, today let's solve what we have on this board. We are asked to solve for the value of x. The first thing we need to do is square both the sides of this equation in order to eliminate these two, okay? So if we do that, we are going to have 2 to the power of x over 2 squared equals x squared, right? Then recall that a to the m raised to the power of n according to the law of exponents is equal to a raised to the power of m times n, okay? So if we multiply x over 2 by 2, this 2 will cancel out the 2. So we are left with 2 raised to the power of x equals x raised to the power of 2. Now let's take natural logarithm of both sides of this equation. So ln of 2 to the power of x is equal to ln of x to the What? 2. All right, now recall that ln(a^x) is equal to x ln a, okay? So this x will come down to multiplying your ln a. We are going to apply this here and here as well. So if we do that, we'll have x times ln 2 equal to this will come down to become 2 times ln x, all right? So let's divide both sides of this equation by x. So if we do that, we have this divided by x, this divided By x. So we cancel this, and we get that ln 2 equals 2 ln x over x, okay? So this is what we have on this side of the board. This is what we have down here. We have that ln 2. Okay, it is equal to 2 ln x over x. But before we continue, if this is the first time you are Upon coming across this channel, please click the subscription button and the notification bell. Okay? So that you'll be notified anytime I upload a new video. Okay, so we need to remove this 2. To do that, let's divide both sides of this equation by 2, okay? So we are going to have Ln 2 divided by 2 equals 2 ln x over 2 times x, right? So we divide these two. So we have ln 2 over 2 equal to ln x over what? x. So by comparison, we will have That x equals 2 by comparison. So this is the first value of our x. Now let's continue. So we establish this relationship that ln 2 divided by 2 is equal to ln x over x. All right, now if we multiply this side of the equation by 2 over 2. So if we say 2 multiplied by ln 2 over 2 multiplied by 2, okay? This is the same thing as ln 2 squared. Right? Using the power rule, 2 times 2, which is 4. And this same thing as ln 4 over 4. So this can also satisfy this particular equation that we established, namely, ln x over x equals ln 4 over 4. Therefore, we can still say that our x is equal to 4, okay? Now what if we multiply this by 3? More than 3? That is 3 multiplied by ln 2 over 2, okay? We are going to have 3 ln 2 over 6. It does not satisfy this equation. That means this is not a solution. 3 is not a value for x. Okay? So if you continue running this test, you'll notice that you're still going to get different solutions that will not rhyme with what we have on the board. Therefore, we can simply say That our x is equal to 2 and our x is equal to 4. Now let's check, okay? So this is the equation. This is the question of 2 to the power of x. Let's use x to be 2. Let's substitute 2 for x. The first value of our x is 2, which is equal to what? 2. So this is going to be 2 to the power of 2. We divide 2 to get 1, which is equal to 2, and 2 is equal to 2. So this is correct. Now what if we substitute our x as 4? We are going to have 2 to the power of 4 over 2 equal to 4, right? So we 2 to the power of 2 is 4. 4 and 2 to the power of 2 is 4, which is equal to 4. So this also satisfies the equation. Thank you for watching. Have a lovely day.