Japanese|Math Olympiad|Exponential math problem

Let's solve for the value of x. What do they have in common? Oh, sorry, x to the fifth to the power of x multiplied by five to the power of x multiplied by five to the power of x now when you a to the power of m multiplied by a to the power of n is equal to a to the power of m Plus n, right? So since they have the same base, we're just going to pick one, which is five. We are going to add the powers, so we have x plus x plus x, and this is equal to 10. All right, so We have five now, x plus x plus x is 3x, which is equal to 10, so at this point we are going to introduce log. Yes, so we take the logarithm of both sides of the equation, and we have log of 5 raised to the power of 3x. is equal to log 10; now, when you have log a to the power of b, this is equal to b multiplied by log a. Alright, so we can simply write our equation as 3x multiplied by log 5 equals log 10. now to get the value of our x we need to isolate anything that is connected to x, right, and to do that we need to divide both sides of the equation by log 5 in order to be left with 3x on one side So we are going to have 3 times log 5 divided by log 5 equals log 10 divided by log 5, all right. Okay, now log 5 is going to cancel log 5, so we have 3x equal to log 10 divided by Log now, um, we have that 3x is equal to. So we can express log 10 as, um, log 5 multiplied by 2. right divided by log 5. Now, when you have log a multiplied by b, this is equal to log a plus log b, okay, all right, so that means that we can rewrite this as 3x is equal to log 5 plus log 2, all divided by log 5. Okay, so we have that 3x is equal to log 5 divided by log 5 plus log 2 divided by log 5, all right, then we have that 3x is equal to this, we can see this to give us one right plus log 2 divided by log 5, all right? Now, if this is the first time you're coming across our channel Please subscribe to the channel and turn on your notification bell—very important, you know. Also, don't miss our amazing videos so you'll be notified every time we upload a new one And don't forget to give us a thumbs up if you're actually enjoying this video Thank you very much. Now recall, or let's remember, this law of logarithm reading that states that log 2 base 5 log a base b is equal to ah, what am I saying? All right, when you have something like this guy, when you have log a divided by log b, this particular law is saying that this is equal to take the logarithm base b, all right, okay, so that simply means that we can rewrite this as what 3x is equal to 1 plus log base 5 of 2. Now, to get the value of x, please stay till the end because we are going to substitute whatever we obtained into the equation to show that it is correct, so please don't miss it that particular amazing video, okay, because that is the most important or the most interesting Aspect of this video, yeah, so we are going to divide 3 by 3, divide by 3, in order to get our x. So you have that x equals 1 plus log base 5 of 2, all divided by 3. Now to the most Interesting aspects of this video—let's check, let's check our solution, the solution we got So let's check now, this is question: 5 to the power of x multiplied by 5 to the power of x. multiplied by 5 to the power of x, so we have 5 to the power of x multiplied by 5 to the is equal to 10. Now, recall that our x—sorry about that—recall that our x is equal to. Let me copy it. From here, 1 plus log base 5 of 2, all divided by 3—this is our x. Okay, so it's time to substitute. so we have 5 to the power of 1 plus log base 5 of 2 divided by 3 okay multiplied by 5 to the power 1 plus log base 5 of 2, divided by 3, all right, multiplied by 5 to the power of 1 plus log base 5 of 2 base 5 everything divided by 3, all right, and this is equal to 10, all right, so we have this. the same base we are going to pick one base which is 5 and we're going to add the powers which will be 3 multiplied by 1 plus log base 5 of 2 divided by 3, okay, all right, is equal to 10. Now when you have a to the power of m times n or a to the power of m raised to the power of n is equal to a to the power of power of m times n, okay, so this is the same thing as 5 multiplied by 3 multiplied by this, and we multiply 3 by this, this is going to divide this, okay, so that means what we now have on the left-hand side the left side of the equation is 5 to the power of 1 plus log base 5 of 2, which is what we have now, is equal to 10 So we are going to apply another law of indices or exponents, which states 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, okay, so that simply means that this is the same thing as 5 to the power of 1 multiplied by 5 to the power of log base 5 of 2 is equal to 10, okay. Now let's apply another law that says that a to the power of log b base a is equal to b, okay? So what's this and this? They are the same thing. This is the answer, so if We apply it here: we have 5, right? 5 to the power of 1 is 5 multiplied by. Then this, since this, and These are the same; the whole of this is equal to 2, which is equal to 10. Okay, now 5 times 2 is 10, and 10. is equal to 2 and 10 is equal to 10, so the left-hand side is equal to the right-hand side. Thank you. Thank you so much for watching, bye!