A beautiful math question for advanced students

hello everyone welcome back to my channel High mathematics and really interesting exponential equation today 4 to the x equal to X and how can you solve this question in this video I'm going to fully explain you how we going to do this correctly and step by step first of all let's rewrite our expression right here so we have four to the power x equal to X let's look at it right here we have exponential function right here we have linear function so right now let's try to expr this in another way first of all let's apply natural loog on both side this is really important important step so we have natur natur log on of 4 to the^ X on the left hand side equal to natural log natural log X uh we apply this natural log because of this x if we apply natural log we all know about this property if we have natural log of a to the power B it is equal to B * natural log natural log a okay this is the main the main thing right here I want you to understand this property this is the main Ro log property so this this B can be written right here before and for this x the same thing X can be written before this natural log 4 so we have x * natural log 4 equal to natural log natural log X okay this is our first step we just apply natural log on both side right now let's try to group and let's try to group our variables on the left hand side and our constant on the on the right hand side uh natural log 4 is a constant so right now let's divide both side by this X as a result we have we have the the next thing we have right here if we divided if we divide by X we can easily cancel this X and this but it appears right here on the right hand side and because of the equal sign we can easily swap position so we can easily write this as natural log X over x equal to natural log natural log 4 okay I hope you understand this step we just swap position we can do this because this is our our equal sign right now we need to know about lambra W function this is really important function and maybe a lot of students don't know about this function a lot of students forget about this function so I try to explain you here in just 10 second this lambra W function first of all imagine you have expression like that you have a time e to the power a where a is wherever you want expression doesn't matter s cosine long complicated expression doesn't matter and if they apply lamb W function of this expression a * e to the^ a it gives us only a okay this is really important moment so we need to create the same a right here the same H right here and the main thing is e in the middle okay so right now let's try to rewrite this expression according to this the let's try to create the same expression according to this lber W function so let's separate this these steps right here and let's rewrite our our our equation so natural log X over overx equal to natural log natural log 4 first of all we should create e because we really need e and first of all let's rewrite this X as e to the power natural log X so we have natural log X over e to the power natural log X this is absolutely the same X as right here in our denominator because e to the power natural log X gives us only X so absolutely the same thing equal to natural log natural log 4 okay what are we going to do next of course we don't need our den numerator so we should bring this uh from our den numerator to our numerator okay so we have natural log X times e to the power minus natural l x because we bring this from from our denominator to our numerator okay e to the power minus natural log x equal to natural log natural log 4 okay right now let's look closely right here we have e it's really great we need to have e right here and right here and right now take a look closely we have natural log X and minus natur log X so we should have the same Expression A and A in our case we have a and minus a and we should multiply both side by minus one because right here it's hard to to cancel this minus we should create minus right here and and then we have absolutely the same expression minus natural log X and minus natural log X right here so let's multiply both side by minus one if you multiplied by-1 we have - natural log x * e ^- natural log X = to minus natural log natural log 4 okay we should multiply both side okay so right here we have minus natural log 4 okay and right now we have minus natural log X and minus natural log X so it looks like it is our a so we can easily apply lambar W function fun so let's let's do this so lambra w function of- natural log x * e^ minus natural log x equal to lambra W function of minus natural log natural log 4 as I told before this complicated expression with lambra W function gives us only a and in our case a equal to minus natural log X because absolutely the same values so as a result we have only minus natural log X natural log x equal to lamb W function of minus natural log natural log 4 okay great right now let's try to find our X because we have minus natural log X let's try to find our our our our X okay so let's do this first of all Let's uh let's rewrite our expression in in a different way first of all let's multiply both side by minus one so we have minus one right here yeah and we have natural log x equal to minus lambra W function of minus natural log natural log 4 okay right now final step right here so let's uh create base of e on both side so right here let's create base of e right here and right here with this base of e we can easily cancel our our natural log so we have only our only our x equal to e to the power e to the power minus lber W function of minus natural log natural log 4 okay so let's simplify this a little bit so let's bring this from our numerator to our denominator so we have X equal to 1 / e to the power lamb W function of minus natural log natural log 4 this is our final answer to our to our question this is a solution to our question but I was wondering how can we find the exact value maybe approximate value to our uh to our question and then I go to internet library and I try to find the exact value of this lambra W function of minus natural log 4 and then I was shocked because lambra function of minus natural log 4 so from here yeah does not exist does not exist does not exist so we can easily say that right here we don't have real Roots because right here this is this expression does not exist so right here this is complex complex Roots right here so this is this uh equation does not give us a real root so X is not X is complex complex roots and I find this complex root for you so X approximately equal approximately equal to minus 0.0 887 plus and imaginary part 1.51 2 I this is our solution but this is a complex complex solution right here so we can easily say that we don't have our real real Roots right here so this is our complex complex complex root and if you look a little bit about graph we can easily see that uh we don't have a point of intersection because 4 to the X this is our exponential function yeah it looks like that for example and our X this is something looks like that y equal to X looks like that so right here we don't have our point of intersection in this moment in this part we don't have point of intersection and moreover in this part our function does not intersect so let's write our answer our answer X x x complex root root and if someone interested in this approximation we have this value and maybe someone interested in approximately answer equal to minus 0.887 + 1. 512 I this is our complex root exact value and this is our solution maybe you if you need this in in your exam so you you should write this write this solution okay this is my explanation to this question really interesting challenge maybe a lot of students think that this is very easy challenge for to x equal to X nothing hard for us but as you can see we should know about lamb W function and we should solve this question step by step this is really interesting moment we can easily go through this question real quick once more first of all nature loog on both side this is really important moment then X goes right here before this natural log form then we group both side we group constant on the right hand side and variables on the left hand side in this moment okay next lamber W function we should know about this lambra W function this is really important function in terms of mass rate here okay so really important function so learn about this function and remember this function this is really important okay applying lamb W function uh according to natural log properties we create base of e right here we multiply it by minus one so as you can see right here this is like a really great algebraic manipulation right here so it's really great you to practice uh this part and final step we find that our lber W function of minus natural log 4 doesn't exist so we don't have real Roots it's really bad but it happens sometimes so we have only only complex root to this really interesting really interesting question and moreover our function does not intersect right here in this part and uh in this part obviously because this is looks like a second part right here this is looks like a third part so it does not intersect moreover they intersect in imaginary part only in these first Parts maybe maybe in a further maybe in undefined values of X they intersect but not in real number and not in real number Roots so this is my explanation to this to this challenge I hope you understand this explanation I hope you learn something new but definitely don't feel bad if you good this wrong if you need help with any of these classes I have a lot of videos on my YouTube channel so if this video is helpful in anyway don't forget to like And subscribe and that definitely helped me out for this particular video so thank you for your time have a great day and see you in the next videos

Channel: Higher Mathematics

Views: 6,076

Keywords: maths, algebra, math

Id: ex_f3BOvNiI

Length: 10min 38sec (638 seconds)

Published: Wed Jul 03 2024