Germany|can you solve for x|Exponential Maths Olympiad

Hi everyone, let's solve for the value of k. Okay, now two is the same thing as three minus one, right? So we are going to replace two here with minus one plus three, so that we have k minus one as well. on this side of the equation, we have two to the power of k minus one plus three, all right then plus two to the power of k minus one equals 15, all right? Now since uh I want us to recall this law of exponents that states that a to the power of m plus n is equal to a to the power of m times a to the power of n, okay, therefore we can rewrite this as what two to the power of k minus one, because I want this to be on its own, all right, multiply by two three to the power of three plus two to the power of k minus one equals 15. Now we have the same. thing here and here, so let's factorize that, so we can write two to the power of k minus one, open brackets when this is divided by this, we have two to the power of three remaining, and when this is divided by this, we have one remaining, which is equal to 15. Okay, now we have two to the power of k minus one multiplied by two to the power of three gives us eight; eight plus one is equal to 15, so two nine multiplied by the quantity of k minus one raised to the power of k equals fifteen; now let's divide both sides of this divided by nine, so now we have divided by nine divided by nine, right? So this is going to cancel. Now we have 2 to the power of k minus 1 equal to 15 divided by 9. Now this 15 divided by now they have common factors, right, because this is two to the power of k minus one is equal This can be written as three multiplied by five, and this can be written as three multiplied by three, so this three can cancel with this three. Now we have two to the power of k minus one is equal to five over three, so this is what we have now. Now I'd like you to watch to the end because I'm I am going to prove that the answer we obtained here is correct by substituting it directly in. The main equation to prove that okay, all right, excuse me. So we are going to introduce AV log at So, log₂(2ᵏ⁻¹) = log(5)/3, okay. Now, when you have log of a to the power of b, this is equal to b multiplied by log a. therefore I can rewrite this as what k minus one multiplied by log two is equal to log five divided by three, okay. Now, to get k minus one on a separate side, I'm going to divide both sides. Divide this equation by log two, so I'm going to have this divided by log two, and this divided by log two. Okay, so I'm going to cancel this, and we have that k minus one is equal to log five divided by three divided by log two. Now, um, I want us to note that when you have log a divided by b; okay, this is equal to log a minus log b; okay, therefore we have k minus one is equal to log 5 minus log 3 divided by log 2, so k minus 1 is equal to we can distribute this denominator to both of them, so it is correct for us to say log five divided by log two minus log three divided by log two, okay please Let me draw my line so that we can continue from here. All right, now this is where we are. that k minus one is equal to, excuse me, okay, so this is where we are, that k minus one is equal to log five divided by log two minus log three divided by log two, this is where we are Now, to have only k on one side, I'm going to transfer this to this side, and when I do that, my minus one becomes plus one, so I'll now have k equal to log five divided by log two minus log three Okay, so this is what we have now. Now, note that when you have log a divided by log b This is equal to log base b of a, therefore we can rewrite this as k is equal to log base two of five minus log base two of three plus one; now this becomes the value of our k All right, now to the most interesting aspect of this video, let's check now the equation is in This form of the equation states two to the power of k plus two plus two to the power of k minus one. is equal to 15; now we are going to substitute this k here, so we have two to the power of log five base two minus log three base two plus one ten plus two okay plus I'm going to write this k is equal to log base two of five minus log base two of three plus one, okay? Then this one has minus one. minus one is equal to 15, okay. Um, let's continue. So from here we can write two to the power of log₂5 - log₂3 + 3 okay + 2^(log₂5) two minus log base two of three, because minus one plus one is zero, that is one minus one, sorry Is zero equal to 15? Note that a to the power of m minus n is equal to a to the power of m. divided by a to the power of n, then similarly 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, all right? So if we look at what we have here Okay, so we have two to the power of this minus this, so it's going to be two to the power of log base 2 of 5 divided by (because of minus 2 to the power of log base 2 of 3) Now, because of this plus, we are going to apply this through which is multiplication. Multiply by two to the power of three, okay? Plus, now this one, we have only minus, yes, so it's going to be 2 raised to the power of log base 2 of 5 divided by 2 raised to the power of log base 2 of 3 is equal to 15. All right, now another thing I want us to note is that when you have a to the power of log base a of b equals b; everything is equal to b; therefore, when you look at this This and this are the same, so everything here is five. Then this and this are the same, everything. Here is three, then two to the power of three is what? Um, two to the power of three is eight. Then, here we have five and here we have three, which equals 15. Now five times eight gives us what? We give ourselves 40. 40 divided by three plus five over three. is equal to 15; now we have the same denominator, we pick one, then 40 plus 5 is 45, which is equal to 15. Now, when three divides four, we have one with a remainder of one. If we divide 15, we have 15. So 15 is equal to 15, and this is correct. Thank you so much for watching. Please click on the Subscribe and turn on the notification bell if this is the first time you've seen it our amazing channel, and please don't forget to give us a thumbs up if you actually enjoyed the video Goodbye