A math olympiad exponential math problem|solve for x.

Back to this place to get 162, okay. Welcome to my channel. I'm going to solve this and then I'll also substitute it back to this to show that what we did is correct; I think that is the the most interesting aspect of this video, so don't miss that aspect, okay? Now we are asked to solve for x, yes, everyone knows that this is the only way, or the only approach to this, is if we introduce Take the logarithm of both sides of the equation. Okay, so we are going to introduce the logarithm to both sides of the equation. equation, so we are going to have log one six two. All right, now when you have log a to the power of b this becomes b multiplied by log a, all right? Now we are going to bring this down, so we are going to have x multiplied by log 27 is equal to log 100 and what 62 all right now when you Look at this; you can see, let's divide both sides of this equation by log 27 divided by log 27. because that is the only way I can get the value of my x now; this we divide, and we have x is equal to log 162 divided by log 27. Now we are going to simplify this further. Yes, we can. just leave this the way it is, let's simplify further, so to do that we have log 162 in terms of log 27 can be written as what x is equal to log 27 multiplied by 6, yes, okay, 27 multiplied by 6 divided by log 27. Now I want us to recall our logarithmic law that states that log of a multiplied by b is equal to log a plus log b, okay, so since we have this here We are going to add them, so we'll have that x is equal to log 27 plus log 6. divided by log 27, all right, so we can write that x is equal to log 27 divided by log 27 plus log 6 divided by log 27. Now when this divides this, we are going to have one, right? That means we can write this as x equals 1 plus log 6 divided by log 27. Now x is equal to one plus, we can further express log 6 as log 3 times 2, right? So we write this as log 3 multiplied by 2, okay? This is multiplication: 3 multiplied by 2, now. divided by log 27 is the same thing as 3 to the power of 3; now we solve that x is equal to one plus now we can write log 3 plus log 2, okay, just like I explained initially, divided by, then this will Come down to give us 3 log 3. Now you can see that we have another two parts of this particular So we have that x is equal to 1 plus log 3 divided by 3 log 3, okay? Now plus log 2. divided by 3 log 3, all right? 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We divide this, so we have one over three. one over three plus this becomes log two, or you can call this one over three multiplied by what log 2 divided by log 3 all right yeah okay now what do we have that x is equal to one over we give us what if we take our lcm to be three I'm going to use uh what is it called the butterfly method: one times three gives us three, plus one times one gives us one, so we are going to end up with four over three, okay, so I will write four over three here, so we have four over three plus one over three log; now when you have log a divided by log b, this is the same thing as log a base b, so we can write this as log base 3 of 2. Now this becomes our equation or the value of x. Now we are going to substitute this back into the particular question to prove that what we obtained Yes, okay, now it's time to prove that what we have is correct. Okay, so all of this is our x. now let's substitute it into this equation to see if we are going to get 162 Okay, so we'll write that 27 to the power of 4 over 3 plus 1 over 3 log the cube root of 2, okay, that everything equals 162, okay. Now recall 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, so we are going to use that the same formula or that same law here, so we are going to have 27 to the power of 4 over 3 plus Sorry, multiplied by 27 to the power of 1 over 3 log 2 base 3 is equal to 162. Okay, now recall that 27 is the same as 3 to the power of 3, so we can write this as 3 to the power of 3 raised to the power of 4 over 3 multiplied by 3 to the power of 3 raised to the power of 1 over 3 log 2 to the power of 3 is equal to 162, okay. All right, now 3: we divide 3; here we have 3 to the power of 4. multiplied by now this 3 also divide this 3 and we have 3 to the power of log 2 base 3 All right, is equal to 162 now. 3 to the power of 4 is 81, so we'll write 81 multiplied by Now, when you have a to the power of log b base a, this is the same thing as what? This is equal to b. to be so we can now write the whole of this as what the whole of this is equal to 2 is equal to 162 Now it's one times two, we get 162, and 162 equals 162. Thank you so much for watching. Have a lovely day