Math Olympiad| Can you solve for x|Exponential question

Hi, let's solve for x. Okay, what they have in common is 7 to the power of x, so let's factor out 7 to the power of x; now, when this is divided by this, we have 1; when this is Divided by this, we have another 1, which is equal to 419. Okay, so what we have is 1 plus 1, which gives us 2. so 2 multiplied by 7 to the power of x is equal to 419; now to remove these two, let's divide both Divide both sides of this equation by 2, okay? So we have 2, or let's just divide it here by 2. divided by 2, so this will cancel out, and we have 7 to the power of x is equal to 2 If we divide 4, we get 2. If we divide 9, we get 4 with a remainder of 1, right? If we divide 10, we get 5, so This 2 4 5 2 4 5, all right. Now at this point, the only option we have is to introduce our log. add to both sides of the equation so we have log 7 to the power of x is equal to log 2 4 5 okay Alright, now when you have log of a to the power of b, this is equal to b multiplied by log a. The most interesting aspect of this video is not even getting the solution; getting the solution is It's very easy now; the most interesting part of this video is where we prove that what we obtained is correct. So don't miss that part, don't miss that part for anything, okay? So that means that from this law I can bring my X down right, so let me bring my X down. That was a joke. That was a joke. Okay, so my x is out, log 7 is equal to log 2 times 4 times 5. All right, now I'm going to divide through by log 7 since I'm looking for the value of x, so that is exactly what I did, and you will notice that this will Cancel this, and we have that x is equal to log base 2 of 45 divided by log base 7. Now let us express 245. In terms of 7, all right, so let's see what it's going to give us. Now, when 24 is divided by 7, we So 35 divided by 7 gives us 5; this simply means that, in terms of 7, 245 is 7 times 35, okay. So we are going to write that here, so our x is equal to log 7 multiplied by 35. Okay, divided by log 7. All right, now x is equal to. Okay, before this, I want us to recall a law. a law in logarithms stating that log(a × b) is equal to log a plus log b Alright, yes, it's a law now. Let's do it here. Let's apply the law here, so you can write that. x is equal to log 7 plus log 35, all divided by log 7. Okay, so let's distribute. Therefore, we can write that x is equal to log 7 divided by log 7 plus log 35. divided by log 7, all right, lest I forget, if this is the first time like this, your first Time to see this amazing channel, oh yeah, click the subscription button and turn on your notifications. Yeah, so you don't miss our amazing videos, like this page— it's always hot, we upload constantly. at least twice every day, can you imagine that? Yes, at least twice every day you have Just use it to refresh your memory, so subscribe to the channel so that whenever we make a post, you'll be notified. You will be notified first, okay? And then give us a thumbs up. It's very important to us. Do you know why? Because it's going to tell the algorithm that yes, this video is actually very Okay, and then it pushes it to even more people, so that more people who really need this for their exam can benefit. I see it, okay, thank you so much. Now we're back to business, therefore. x is this, we divide this to give us one, so we have one plus log 35 divided by log 7. Now, if you look at this, notice that we can still express log 35 in terms of log 7, right? so we can write that x is equal to 1 plus log base 7 of 5 divided by log base 7, so x is 1 plus log 7 plus log 5, remember that I've explained this before, yes, so we have multiplication. add divided by log 7, so we can say that x is 1 plus log 7 divided by log 7 plus log 5 divided by log 7; now let me clean this up, it's actually bothering me, all right So we can write that x equals 1 plus 1 because when this divides itself, we get one, right? So 1 plus 1 plus log 5 divided by log 7; so we can write that x is equal to 1 plus 1, which gives us 2. plus log 5 divided by log 7, I don't know if you can still see this, so let me just quickly do that Here we are close to our solution, please stay put. So we have that x is equal to 2 plus I'm just I'm going to repeat what I wrote there in case you can't see it all right, so this is What we have over there, therefore, x is 2 plus log base 7 of 5, yes, because when you have log a divided by log b, this is equal to log a base b, okay, yeah, so that's why I wrote this as log 5. Base 7? Yes, let's get the most interesting aspect of this video. Now let's check, okay, let's check, let's check, let's check. The video says, the question says. 490 Okay, now our x is this 2 plus log base 7 of 5, right? So that means we have 7 to the power of 2 plus log base 7 of 5 plus 7 to the power of 2 plus log base 7 of 5 equals 490 now we have the same thing into two places, so it's the same thing as two times that same. Okay, so we'll write this is equal to 2 multiplied by 7 squared plus log base 7 of 5, and that is equal to 490, let me even put a question mark— is it really equal to 490? Okay, so this is two brackets; when you have that when you have a plus b as your exponent is the same thing as the base raised to the first power multiplied by the base raised to the second power, I've already explained that before, so this 7 squared multiplied by 7 to the power of log base 7 of 5 equals 490 Okay, so when you have, let me say, something here, when you have log, oh sorry, when you have a to the power of log base a of b; this and this are the same; okay, this is equal to b; therefore We can write here as two brackets: 7 squared is 49, multiplied by that means since we have The same thing here; this is the answer for here is equal to 490, okay, all right. Let me switch to clean here. Now we have two brackets: 49 times 5. What do we get? 5 times 9 is 45, right? Write down 5, carry 4. 5 times 4—sorry, 5 times 9 is 45. We write down 5 and carry 4. 5 times 4 is 20, 20 plus 4 is 24, so this is what we have, okay, is equal to 490. Now 2 times 2 is 4. Sorry, what am I doing? 2 times 5 is 10, you write down 0 and carry 1. 2 times 4 is 8, plus 1 is 9. 2 times 2 is 4, so we have 490 is equal to 490; the left-hand side is equal to the right-hand side. Thank you for watching