Germany| Math Olympiad| Exponential math problem| Can you solve for a

Welcome, let's solve for a recall that the law of exponents a to the power of m plus n Okay, is equal to a to the power of m multiplied by a to the power of n, all right, so but also recall that plus one can be written as two minus one, so we can rewrite this as 3a minus 1 plus 2 plus 3a minus 1 equals 12, so you can see that we have 3a minus 1 here. and 3a minus 1 here, but before that, let's apply this particular law. So when we apply it, you 3^(a - 1) × 3^(2 + a) minus 1 equals 12 now let's factor out 3 to the power of a minus 1 so when we do that we will have 3 to the power of a minus 1, sorry, this minus 1, right, open bracket, when this is divided by this, we are left with 3 squared plus, when this is divided by this, we are left with 1 is equal to 12, so we have that 3a minus 1, um, 3 squared plus this is the same as 9 plus 1. is equal to 12. 3 to the power of a minus 1, bracket 10 is equal to 12. Now let's divide divide both sides of this equation by 10, so we have divided by 10 divided by 10, so this is going To cancel this, we have three brackets: a minus 1 is equal to 12 divided by 10. Okay, so we continue. we have 3 to the power of a minus 1 is equal to when um 2 divides 12 we have 6 when it divides 10 we have 5, so we can write this as 6 divided by 5. Now we are going to introduce log to both. On the sides of this equation, I would like you to watch to the end and see how we substituted what we obtained. into the main equation to prove that it is correct. Okay, so we are going to introduce log 3 to the is equal to log 6 divided by 5. Now, when you have log a to the power of b This is equal to b multiplied by log a. All right, all right, so that means we can write this as a. -1 multiplied by log 3 is equal to log 6 divided by 5, okay, then we have a minus 1 log 3 is equal to when you have okay before that, let me not confuse you, so let's divide. both sides of this equation by log 3, okay, so divide that by log 3, then we have log 6 divided 5 divided by log 3, okay, so log 3 will cancel log 3, and we have minus 1 is equal to log 6. divided by 5 divided by log 3, okay, if this is the first time you've seen this channel, please Subscribe to the channel and click on the notification bell so that you'll be notified anytime I upload. A new video, so you don't miss our amazing videos, okay? And give us a thumbs up if you're actually Enjoying this video, thank you so much. Okay, so let's continue. Now recall another law of indices. or exponents that states that log a divided by b, okay, sorry, you can't see that, so it states that log of a divided by b is equal to log a minus log b, okay? All right, so that means that we can rewrite this as what a minus 1 is equal to log 6 minus log 5 divided by log 3, so we have minus 1 equals log 6 divided by log 3 minus log 5 divided by log 3, okay, all right, so let's continue. We have minus 1 is equal to log 2 multiplied by 3. divided by log 3, since 2 times 3 gives us 6, right, minus log 5 divided by log 3, so we have a minus 1 is equal to when you have log a times b, this is equal to log. Oh, you can't even see that. Alright, so when you have log a times b, this is equal to log a plus log b, okay. so we are going to rewrite this one nowpolylog 2 plus log 3 divided by log 3 minus log 5 divided by log 3, so we have minus 1 is equal to log 2 divided by log 3 plus log 3 divided by log 3 minus log 5 divided by log 3, all right, so we have a minus 1 is equal to log 3 divided by log 3 we get 1 plus log 2 divided by log 3, okay, minus log 5 divided by log 3, okay, but we have a minus here; we have minus 1 here, so let's move it to this side, and when We do that; we have a minus 1. Is it sorry, a? Because we are going to transfer this, so we are going to Add 1 to this side and remove it from here so that a equals 1 plus 1, then plus log 2 divided by log 3 minus log 5 divided by log 3. Okay, so 1 plus 1 will give us 2. 1 plus 1 will give us 2. so this is what we'll have right now, so we now have a is equal to 1 plus 1 plus log 2 divided by log 3 minus log 5 divided by log 3, so this is what we have now. We continue; we can see that a is equal to 2 plus log 2 divided by log 3 minus log 5 divided by log 3, okay, but when you have log something over log something; once the log readings have the same base, you can actually Write it in such a way that a can be expressed as 2 plus log base 3 of 2, minus log base 3 of 5. This is the value of our A-fine now let's get to the most interesting aspects of this video which is what, let's check our results because we are told that 3 to the power of a plus 1 plus 3 to the power of a minus 1 is equal to 12 this is the question okay this is the question Now let's check when you look at it, we are now going to substitute our a, so we have 3 to the 2 to the power of 2 plus log base 3 of 2 minus log base 3 of 5, that is for this one, then plus 1 plus 1 plus Let's go for this: 3 to the power of 2 plus log base 3 of 2 minus log base 3 of 5. Okay, then this one has minus 1 minus 1, all right. Now everything is equal to 12. Is it now? Let's continue. So we have 2 plus 1 we give us 3 happy yes so that means we can rewrite this side as 3 to the power of 3 plus log base 3 of 2 minus log base 3 of 5 plus, for this side, 2 minus 1, which gives us 2, so we can write 3 to the power of 2 minus 1 gives us 1, so we can write 3 to the power of 1 plus log base 3 of 2 minus log base 3 of 5 equals 12, so we can write that 3 to the power of when you have a to the power of 3, oh sorry, when you have, say, x to the power of This is equal to x raised to the power of a multiplied by x raised to the power of b. Okay, so let us remember. that now when you have x to the power of a minus b is going to be x to the power of a divided by x to the power of b, so we are going to relate this to what we have here now, so we can now write this as what 3 to the power of 3 multiplied by 3 to the power of log base 3 of 2 divided by of this division, 3 to the power of log base 3 of 5, okay, all right, plus the same thing is what we are going to do to this side so we have 3 to the power of 1 multiplied by 3 to the power of log base 3 of 2 divided by 3 to the power of log base 3 of 5 Okay, so don't get confused, all right? So this is equal to solve now 3 to the power of 3 we give us 27 right multiplied by now when you have log when you have a let me write this Now when you have the logarithm of b to the base a, this is equal to b; everything here is equal. To be okay, so that simply means that we can write this as what, okay? So that simply means that we can write this as 3 to the power of 3 giving us 27 multiplied by now because of the base and this They are the same, so we have to stay here divided by what now—the same thing is happening젴 Same base, same base, so the only thing remaining here is 5, okay, plus then we have 3. to the power of 1, which is 3 multiplied by the same thing, only 2 will remain here, 2 divided by The only thing that will remain here is 5 plus 5 equals 12, okay. All right, now 27 divided by We give us 54, so we have 54 divided by 5 plus 3 times 2, which is 6, divided by 5, is equal to 12, now we have the same denominator, so we can just add the numerators: 54 plus 6 gives us 60, divided divided by 5 equals 12; now, 60 divided by 5 gives us 12, and 12 equals 12, so we can see that The left-hand side is equal to the right-hand side. Thank you so much for watching. Bye.