Hardest exponential math question | Harvard University|solve for x

Sorry, my voice is gone, so let's use this exponential law that states that a to the power of x minus m, something like this, that is equal to a to the power of x divided by a to the power of m, so if we should relate this to what we have here so that simply means this can be written as two to the power of x divided by two to the power of one is equal to two x. Now, to get the value of x, we need to bring anything that move x to one side and then leave the constant on the other side. To do that, first let's If we cross multiply, we are going to have two to the power of one multiplied by two x. is equal to 2 raised to the power of x, so here we have four times this, which is 2, right? Two times two gives us four, so we have four x is equal to two to the power of x. Excuse me, now um let divide the equation by 2 to the power of x. Remember, I said I want to bring x together, okay? So let's divide both sides by 2 to the power of x, so if we do that, we'll have 4x divided by 2 to The power of x is equal to two to the power of x divided by two to the power of x, so we are going to have 4x divided by 2 to the power of x equal to 1, all right? Now this is the same thing as 4x multiplied by 1 over 2 to the power of x right is equal to 1. Now recall that 1 over x is equal to x to the power of minus 1, so 1 over 2 to the power of x is equal to 2 to the power of minus x. x to the power of 2 to the power of minus x, sorry, so if we should substitute this into this place, so we are going to have 4x multiplied by 2 to the power of minus x is equal to 1 now we need to divide both sides of this equation by 4 remember I want all anything Regarding x to be one side, okay, so let's divide this by 4 and divide this by 4, so we are going to have this 4, we cancel the 4, so you have x multiplied by 2 to the power of minus x is equal to 1 over 4; now at those points, I cannot solve this using any algebra knowledge, okay Rather, I can only solve this using the lambda w function. Okay, now what is the lambda w function? The lambda w function states that if you have something like this, a times e to the power of a. I will introduce the lambda w function to it, and the answer to this is a, all right, so that simply. means that I can manipulate everything here to be in this form, the more we have e and whatever that is multiplying e and this raised to the power that is also raised to the power should be the same Is it all right if I manipulate this to be in this form and this side to be in this form? We are definitely going to get our value, all right, so in order to do that, this is what we are going to do, we are going to introduce a negative sign to both sides of this equation; I want you to look At this, this is 2 to the power of minus x. Remember my power, and whatever I'm multiplying, which should be the same thing, so minus x; in order to get minus x here, I need to multiply this side by minus x. minus one, so I also need to multiply this side by minus one to balance it. So if we multiply both sides by minus one, we are going to have minus x multiplied by two to the power of minus x is equal to minus one multiplied by one over four. Now, if you look at this, we have the same thing Here and here, but we don't have E. Remember that you must have E. All right, so in order to get this record that e to the power of ln a is equal to a, so that means that e to the power of ln 2 will Still give me these two so I'll replace these two with e to the power of ln 2 so I won't have a minus x times e to the power of ln 2; everything here is to raise the power of this minus x is equal to minus One multiplied by one over four, right? Okay, so if we expand this bracket, we are going to have minus x. e to the power of minus x ln 2 right is equal to minus one multiplied by one over four now notice that our power now is minus x times ln 2, right? Meanwhile, we want whatever that is here to also be here. Now the difference between this and this is ln 2, so I'm going to multiply this side by ln 2, so I also multiply this side by ln 2 to balance it off, okay, so we are going to have ln 2 multiplied by This minus x times e to the power of minus x times ln 2 is equal to minus one multiplied by one over four. So when this multiplies this, we now have minus x times ln 2 times e to the power of minus x. ln 2 is equal to one over four multiplied by minus one ln 2. Now, if this is the first time If you're coming across this channel, please click the subscription button and turn on your notification bell so you don't miss our amazing videos—we upload videos daily So please tune in at your convenience to watch our new uploads, okay? All right, please give us a thumbs up if you actually enjoyed this video, thank you very much. Now when you look at them On the left-hand side, you'll notice that what we have here is the same as what we have as power. So this is the right time to introduce the lambda w function for the left-hand side. but when you look at the right-hand side, notice that it's still disorganized, so we still need so that whatever we have multiplying our e should be the power Of our all right, so we can do that here, yes, it's possible. Now, how do we go about this? So this is equal to one over four; now, when you have minus, when you have anything multiplying your ln all Right, you can use it as power using the power rule, right? So we can write this as ln 2 raised to the power of minus one. All right, now this can be written as one over four ln. Now two to the power minus one is the same thing as half, all right? Okay, so permit me to draw this line now. This is where we have minus x times ln two times e to the power of minus x times ln two equals one over four ln half so this is what we have now all we are trying to do is to manipulate this side to have the same thing, just like what we have here, all right, so we have that this is equal to one over four now we can decide to multiply this by four before we have not changed anything right so we have 1 over 2, so this is equal to 1 over 4 times 4 is 16 multiplied by 4 One over two, right? So this is equal to one over sixteen. So this four can become the power. of this right, so we have ln one over two raised to the power of four, okay, so this is equal to one over sixteen. ln one to the power of four is still one, and two to the power of four is sixteen. Now at this point, let's introduce our E. All right, so if we introduce our E, we have, remember that e to the power of ln one over sixteen is the same as one over sixteen. We have this; we have replaced this with this, so we still have ln one over sixteen here, all right. so this is equal to; now remember that the lambda w function is in this form, is in this form, so our The e is supposed to be in the middle, all right, so that's where we are going to bring this e. to come to this side and then take this to this side, so we have ln one over sixteen e to the power of ln one over sixteen again, so you can see that now we have the same thing here and as a power, so Let me bring down the other one, this one, so that means that what we have now is minus x ln two raised to the power of e minus x times ln two, okay, is equal to this. Now let's introduce the lambda w function for both sides, so we have the lambda w function of minus x ln 2 e to the The power of minus x times ln two is equal to the lambda w function of ln one over sixteen times e to the power In one over sixteen, all right. Now recall that what this will give us is this, and what this will give us is this, so we are going to have minus x times ln two is equal to ln one over sixteen, okay Perfect, right? This is interesting. Yes, now look at what we have: we have minus x times ln two is equal to Now, one over sixteen is the same as sixteen to the power of minus one, so you have minus x times ln Two is equal to excuse me, so this is ln now sixteen, the same thing as two to the power of four. and we have this raised to the power of minus one, all right, so minus x ln two is equal to ln two to the power of four times minus one is minus four, okay, so we have minus x times ln two is equal to. Now, using the power rule. This minus four will come down to multiply ln two, so we have ln two here and ln two here. Let's Divide both sides of this equation by ln 2, so we have divided by ln 2 divided by ln 2. Now ln 2 we cancel ln 2, we have minus x is equal to ln 2, we also cancel ln 2 is minus Now to get the value of x, let's divide both sides of this equation by minus one divided by minus one divided by minus one, so minus we cancel minus, we have x, and minus we cancel minus, we have Now let's verify, okay, let's verify this my voice. So the question says two to the power of x minus one is equal to two x; now we have our x equal to four. All right, so we have two to the four minus one equals two times four, so we have two to the power of four minus one is three equals two times four equals eight; now two to the power of three is two times two times two is equal to four; now two times two is four, four times two is eight, so we have Eight is equal to eight, so this proves that what we solved is correct. Thank you so much for watching. Goodbye