How To Solve Exponential Equation x^5=9^x || Solving Exponential Equations.
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- Опубліковано 10 лют 2025
- How to solve exponential equation of this kind is rhetorical but I will guide you on simple ways you can solve this kind of challenging exponential equations with easy.
In this video tutorial, I will explain to you on how to apply the Lambert W function to solving challenging exponential equations like this.
Watch from beginning to the end without skipping any parts for the full details.
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I am a math student(pursuing under graduate degree in India) from india and i just watched your video by UA-cam suggestion and I like you way to solving problem with explain in very eazy language and I like it thank you please made more videos like this for improving students mathematical knowledge. Thank you🎉❤
Love the clever way to solve problem. However, the solution’s argument for the W Lambert function (-Ln (9)/5) is outside the domain of such function [-1/e, ∞). The curve X^5 - 9^x does not intersect the X axis, the closest value (y=-0.345) occurs when x=0.589.
2222232-81=
000000032
-49
Rafael is absolutely right - given that (-ln(9/5) < -1/e, the conclusion of this exercise should have been to prove at the end that there are no real solutions to above given equation. In relation to that, It would have been useful to theoretically explain this outcome by sketching W(x) function and concentrating particularly on the meaning of the point -1/e in order to understand the concept behind the proof.
A complex solution is still a solution!!!
Its first time I saw the lambert W Function, when I study Calculus I and II, my lecture never taught this Lambert W Functions. I'm very blinde about this them... Please make more video especially about this Lambert W Functions. For your respect, thanks very much.
You are really great mathematics and very good you are great preacher
Nice solution
Thank you teacher you are so good for math.
Lambert function😮 is new for me.
yea for me too
I like wrapping my mind around these Olympiad problems. They are most cleverly put together. They're designed to make us think outside the box to come up with a simple, elegant approach to the problem at hand.
Thank you very much, professor. This was the first time I saw the Lambert function used to solve an equation. This is amazing! And it's nice to learn new things!
Nice
🎉🎉🎉🎉
Спасибо, профессор, Вы великолепны!
🎉well said sir
The domain of the W function of a real number is [-1/e, ∞). The argument (-ln 9)/5 is less than -1/e. Consequently, W((-ln 9)/5) does not have a real number value. The two curves do not intersect in the real number plane, as can be confirmed by plotting them using the online graphing app Desmos.
- ln9/5 is greater than --1/e because the interval semi close interval.. So, -- 1/ e is real. Negative number is real, not imaginary.
-0.881024 + 0.59038 i. Is the approximate answer to the Lambert W function.
@@syedmdabid7191
Product logarithm is multivalued function in real number.
W0: [-1/e, ∞) -> Real number
W1: [-1/e, ∞) -> Real number
Dom W= [-1/e, ∞)
3 > e > 2.5
ln x ↗
ln 9 > ln (e²) = 2 >5/e
(ln 9)/5 > 1/e
-(ln 9)/5 < -1/e
-(ln 9)/5 ∉ [-1/e, ∞)
W [-(ln 9)/5] ∉ Real number
=> The solution can only be in complex numbers; there is no solution in real numbers.
Oh but you see, there actually are multiple branches of the lambert function, spanning the entire complex plane and real line :>
Oh but you see, there actually are multiple branches of the lambert function, spanning the entire complex plane and real line :>
love from sri lanka....😍
Thanks a million sir
very good algebric problem
Incredible professor, thanks
You are most welcome sir.
And thank you for thanking us and appreciating our little effort sir.
We all here love you ❤️❤️💕💕💖💖
very good
Thank you my good friend
Incrível!
Nice explanation sir
Many thanks to you for the acknowledgement sir.
Much love from everyone @OnlinemathsTV ❤️❤️😍😍💕💕💖
I first came accross this type of problem when I was trying to answer a question regarding compound interest vs simple interest : ax = b^x where "a" and "b" are known quantities.
I never went to olympiad. but as sweet you solve this, I really enjoy and occuping with it. Many thanks dear bro!
This is great approach master. I love this, and i have learnt some thing new sir. Thanks master 🎉🎉❤❤
Happy to hear that!
Good one
You really love math and teaching!!!
Nice video
Thanks for watching and finding some values in our video sir.
Much love master.
Nice sir
Great work.From India.
great approach but.....😍😍😍😍😍😍
What's wrong with the content sir? We are very open to corrections and criticism.
Great Lambert function (W)
Thanks for watching and commenting sir.
Much love from all of us @onlinemathstv to you sir 💕💕💖💖🙏🙏.
Nice explanation professor thanks 🙏🙏🙏
I didn't get past the 2nd line, you're good Sir, very good.
Please make more such videos on harder algebra.
Thank you professor
Brilliant work again thanks so much teacher
Thanks for the encouragement sir.
Much love
Hi sir
Please I need to understand as to how you immediately introduced logs on this without first tackling in normal way as others in this class.
i attempted another solution using the same reasoning:
define W: W^5 = 9^W. x=W QED
asnwer=3 isit hmm no math matter like isit 😂🤣😆😅
The first time I understand the Lambert functions
I understood, THANKS A LOT
You are welcome, and we are glad u gained some values from this video tutorial sir.
Much love 💖💖💖
Great! But inconclusive.
Thank you for the video!! Turn my maths fear and anxiety earn in the past 15 years from a teacher I had…to an opening in mathematics again
Its a pleasure serving you in this regards and we are glad you gained some values from this video tutorial sir. We will do our best possible to serve you and others better in this endeavor.
Maximum respect sir...🙋🙋
Thank you Sir, your response is so overwhelming. Thank you
very nice approach
Glad you liked it and thanks for watching and commenting sir.
Much love 💖💖💕💕💕
Well, when I first saw this I thought “there’s no way 9^x ever meets x^5 , right? Cause a power function is always above the x-axis…and there’s no way for any x^a (a>0) to catch it since such functions flatten out from 0 to 1…so…what is this guy smoking?”
Perhaps I’m missing something? Like a good hit. :p
三八弟
Am smoking maths hahaha...🤣🤣😂
@@onlineMathsTV Oh I see your solution must be an element of the Complex numbers...ok..haha...This is all new math for me, but I get it now. ps You can google "Wolfram Alpha" and then enter "productlog(A)" ...which is equal to W(A) pps Can I offer a little unsolicited advice? Let your audience know the domain of your x whenever you introduce a function. Also, point out that W(A) will only have real number solutions if A is greater than 1/(-e).
Спасибо учитель,я не понемаю твое слова,но понемаю твое решения задач,я блогадарень тебя,очень хочу с тобой обшатся,я из УЗБЕКИСТАНа,знешли ты такое государства
Aproksimativno resenje je kompleksan broj x=2.004857+i×1.584805
I only watched the video because I couldn't solve it and I realised that it probably has no solution because the graphs of Y = x^5 and y = 9^x do not intersect. I thought I will be impressed if he can solve it and tell us the solution but he didn't give us any solution at all, he simply played with symbols. He should have been honest from the beginning and said this actually has no solution.
@averycleverman But fewer people would watch a video called "I cannot solve an equation that has no solutions."
Well it has no real solutions but It actually 5 imaginary solutions and this is one of them
17 years old account 😮woww that's great 👍 🎉
There are no real solutions, but they are complex.
Does the answer approve x^5= 9^× ?
This is the real solution and no doubt about it x = -(5 W_n(2/5 (-1)^(3/5) log(3)))/(2 log(3)), n element Z or
One of the solution of x
x≈-2.2756
Great work, sir it was vey hard challenge, bt any way thanks.
Thank you for watching and appreciating our small effort so far.
We love you sir❤️❤️💕💕
it is simple
Sure master, it shows you are the master here and you are good at what you do sir.
Respect👍👍👍.
Great! Thanks for the rigorous approach! Greetings from Italy!👍👍👍👏👏👏👏
You are most welcome sir, and thank you for watching our video tutorial and leaving a comment behind.
We love 💖💖❤️❤️
Sir,please express of W
But which exam contains like this type of questions please tell me because I want to know?,
Professor, what does W mean in solving multiple equations?
Lambert function
답이 없는 문제임.엉터리 해법
@@조광일-h5i 본인이 모르면 답이 없는 것인가요? lambert function이라고 위에 답을 주었는데도 인터넷도 한 번 안 찾아보고 엉터리라고 말하다니..같은 한국인인 제가 부끄럽네요.
@@조광일-h5i학무위키라도 읽어라
@@태권정-z8yㅋㅋㅋㅋ 바보야
It's beyond my ability to understand
Какой приятный парень.
Но я не понял решения в конце - что за функцию он ввел, когда приравнял А на Е в степени А к просто А? Чему же равен Х? Что за функция в решении?
Interesting exercise, rigorous approach...
Thanks ! Congrats ! Keep it up ! Good luck with your work !
😍😍😍 Thanks so much for this wonderful comment you dropped here.
@@onlineMathsTVut lambda ₩ 👍؟؟????🦧🦧
@@onlineMathsTVs this college level math? If so what class is it? I never seen this in high school
Hi❤❤❤
Prof. Are u from Nigeria?
Just found that the general solution for "xᴬ = Bˣ" is:
x = - W(ln b/a) / (ln b/a)
Obviously I haven't really paid attention to W domain subjects.
does x = e ^ -W(- ln9/5) valid?
Hy.
This equation don't have real solution.
The graphic of f(x)=x^5-9^× is strictly negative. Tx
Bravo👍👍👍
You are the master here because you are good at what you do🙏🙏🙏
Respect👌👌👌
Very impressive usage for Lambert W function, but if you do not use a few more explanations about "Productlog function", I think many of your audiences will not eventually reach the final point.
so what is the value in decimal form? I want to substitute it to double check.
You can get that from your Wolfram Alpha calculator sir.
La wolframalpha no da un valor numérico en es te caso
@@gerar2158 claro que sirve: 2.0049 - 1.3435 i
Nice explanation and thanks for this video Jakes.
Thanks for appreciating our little effort sir
Much love from Onlinemathstv 💖💖💖💕💕💕
Base =9
Power : 5
So, what's is the numerical value of x that you sought now?
x ≈ 2.0049 - 1.3435 i
Compute the Integral x^2024/(x^2+1)^2023 dx please.
It's interesting, that in according a formula
e^nW (x) = [x/W(x)]^n:
n = -1 and
e^-1 W(-ln 9/5) = [(-ln 9/5)/W(-ln 9/5)]^-1 =
W(-ln 9/5)/(-ln 9/5)
I asked Chatgpt, and the answer is 2.478.
Teacher what does mean w in this equation?
Hmm, he divided "mindlessly" by x on both sides of the equation (video 3:10 ), so he basically assumed x to always be positive. I guess you could do that if you consider that logarithm functions are defined for positive values of x only, am I right?
وظائف الوغاريتم غير محدودة في مجموعة الاعداد المركبة يا صديقي
you can check that 0 isnt in question. and the fact that X can be negative is not to be underminded. in fact, x has a possibility to be negative, as there are infinite solutions for the equation. Since we are solving a complex equation, its not necessary to check if x
X^√5=√9 1√^√1. 3^3 (x+3x-3)
I think am seeing a unique approach unfolding here.
Can you kindly enlighten Onlinemathstv on this approach, please sir?
I was going to say that... (who's Oliver?)
I can prove that x^5
When the question asked for value of x, it didn't ask for real value only
What is the numerical value of W(-ln9/5? Which is the exponent of e?
x = e^(2ln3/5) in the real plane
W₀(-ln9/5) ≈ -0.88102 + 0.59038 i
W₋₁(-ln9/5) ≈ -0.88102 - 0.59038 i
W₁(-ln9/5) ≈ -2.9052 + 7.4837 i
W₋₂(-ln9/5) ≈ -2.9052 - 7.4837 i
...
Would you please make a video on Lambert Function, professor!?
Does the equation have a solution?
Yes, but only complex.
.x=3
X=9
I got that the risult is approximatively 0,34+0,23i 😞 what does that means? That the solution is irrational?
We really appreciate it.
2.4 (x+2x-4)
I do not understand this sir, kindly rephrase your point sir/ma.
But how could you find the numerical value of W-n ( -- ln 9/5)??? For this, you must have to apply Newton Raphson method / graphic method.
But won't it have multiple solutions? As the input is negative?
I'm pretty sure there is no solution to this equation since if u plot the curve of x^5 and 9^x, they don't have any intersecting points!
🇩🇿🇩🇿 شكرا
So what is x that solves the equation? Give us the value.
Plotting f(x) = x^5 - 9^x I have satisfied myself that f(x) is always < 0, with the maximum of about -0.34 occurring near x = -0.6, i.e. there is no real solution to f(x) = 0.
Am I wrong?
no, you are correct. There is no real solution. But, for some equation of the "n^x = x^n" sort, it is very useful to keep the W in mind. There are infinite solutions in the complex plane, meaning there is no single solution( because the lambert function takes in a number for its power, but in this case the number is absent, meaning it can be anything from the real plane.)for the equation, rather a set of infinite ones. there is a total of 5 formulas for solutions, hence 5 infinite sets of solutions(not counting the lamber power). I can give you one right now, approx. 7.92822 -31.61131 i, where i=√-1.
So what is X in numerical values?
Я нарисовал график в geogebra и там эти функции не пересекаются😮
Because the equation has no real solutions. Only complex ones.
This is only one of the 5 solutions. Can you share the other 4 solutions? I listed all 5 solutions using log function below in the reply to my comment. If you can show the full solutions that will be helpful.
W(-ln9 / 5 ) is a real number itself?
Where did the W on the left go? 🤔
This great but I think this math is beyond my level. Anyway, thanks for dropping this nice video sir.
You are most welcome
Answer 9
You have not yet checked whether your solution is correct or not
Dear professor! What is ld function?
But the image of 9^x does not intersect with the image of x^5
x is a complex number; see Demetrius above !
See answer of Ratko Dinic.
The Lambert function of a negative value (-log(9)/5 < 0 always) will result a not real value. So, there is no x € |R that satisfy x⁵ = 9^x.
but what is the value of λ？
Now. All you have to do is to solve the exponential of the Lambert function of a negative natural log that is outside of the valid range for real roots. Ok. trivial just solve for the complex root(s) of .... Btw - I have a bridge for sale at a very reasonable price. The Lambert function is just a dodge to claiming to solve an equation that is not solved at all. Well of course, unless you want to provide an analytical equation for the derivation of solutions of general Lambert equations.
How to evaluate W(- ln9/5)?
Use the Wolfram Alpha calculator, it will give you the numerical value
Sir; can you explane that rules name 6:32
how can i find that
So this is where we landed ? Just like leaving Egypt and still faced with the Red Sea
🤣🤣🤣🤣😂😂😂🤣 you make me laf uncontrollably @Aloysiusdaniel.
You can get the numerical values from Wolfram Alpha calculator at this point sir.