Germany| A Nice math Olympiad exponential question.

Hello, it's another amazing math olympiad question. Join me, let's solve it together. Um, let's transfer. subtract 2a to this side, so we are asked to solve for the value of a, so 2a plus 2a is equal to 40 Now, what do they have in common? They have 2a in common, right? Let's factor that out. 2a we open our bracket, 2a divided by 2a gives us 1, and 2a divided by 2a gives us 1. and this is equal to 40, so we have 2a multiplied by 2; this place is equal to 40. Now, to simplify this, let's divide everything by 2. Okay, so if we divide everything by 2, we will have divided by 2 divided by 2, so these two, we divide this term, we have that 2 to the power of a is equal to 20, all right? Now at this point, we have to introduce log, okay? So let's introduce it. Take the logarithm of both sides of the equation, so log of 2 to the power of a equals log of 20, all right. Now, according to the law of logarithms, log a raised to the power of b can be expressed as b. multiplied by log a, okay, so let's go ahead and substitute that, please. I would like it to stay. to the end of this particular session or this particular solution that we are trying to find Please stay until the end of the video, because this is the solution we are going to get The amazing part of it is where we prove that what we're solving is correct, so I advise you to stay. Alright, thank you so much. Now this is what we have—we are going to have a log 2. Right is equal to log 20. All right, now we are looking for a, so let's divide both sides of this. and if we do that we have log 2 divided by log 2 is equal to log 20 divided by log 2, all right, now log 2 is going to cancel log 2, and we have that a is equal to log 20. divided by log 2, but log 20 can be expressed as what log 4 multiplied by 5, right? 4 multiplied by So let's go ahead and do that. So we have that a is equal to log 4 multiplied by 5. divided by log 2,_grp so we have that a is equal to log 4 plus log 5 divided by log 2, yes, because according to the law of logarithms, log a b means log a multiplied by Because b equals log a plus log b, that's why I expressed it in this form, okay. So I want to change position now. All right, now we have that a is equal to log 4 divided by log 2 right plus log 5 divided by log 2 all right so a is equal to log 4 can be expressed as 2 to the power of 2 divided by log 2 plus log 5 divided by log 2 now our a is equal to using the power rule as we did before, we are going to transfer this to this place and we have 2 multiplied by log 2 divided by log 2 plus log, now I want us to note that log a divided by log b equals log a base b, okay, so let us, um, implement that here. So this becomes log base 2 of 5. Now, divide log 2, and we have that a is equal to 2 plus what? log base 2 of 5, now, as I said, the interesting part is how we prove that this answer is correct. if this is the first time you are coming across this channel, please don't hesitate to click on the subscription button and turn on your notification bell, very important, yes, so that you'll be notified Whenever I upload a new video, I won't miss lovely sessions like this, okay? So please don't Don't forget to like, comment, and share—that's where the algorithm pushes my video far away or to more. people who need it, okay, thank you so much. All right, now this is our solution, let's Prove that it's right, all right, so I'm going to clean this because I feel more comfortable staying. Like this, because I'm right-handed, okay, so um, we have that our a is 2 plus log base 2 of 5. Alright, now the equation says what 2 raised to the power—I'm going to use the second one另一种说法：现在方程说的是2的幂——我将使用第二个。 2 to the power of a plus 2 to the power of a equals 40, so let's prove that now we have 2 to the power of that is 2 plus log base 2 of 5 plus 2 to the power of 2 plus log base 2 of 5 40 Okay, now we have 2 to the power of 2 multiplied by 2 to the power of log base 2 of 5. okay now why is that because when you have a to the power of m plus n it is equal to a to the power of m times a to the power of n m multiplied by a to the power of n, all right, so that is why I had to separate this this way and similarly we are going to have the same thing here 2 to the power of 2 multiplied by 2 to the the logarithm of 5 to the base 2 is equal to 14, okay? Now, 2 to the power of 2 is 4, right? Multiplied by Now when you have a raised to the power of log base a of b, this is equal to b. This is Equal to b. All right, that means that everything we have here is equal to 5, right? Okay, plus 2 to. The power of 2 is 4; multiplied by everything here, 5 is equal to 40. Now, 4 times 5 is 20. 4 times 5 is 20, and 20 is equal to 40. Now we have that 40 is equal to 40. Amazing, right?