For questions related to the Lambert-W or product log function. This is the inverse function of $f(z) = ze^z$.

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0
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2answers
62 views

How do you solve x^2 = log^2(x)

I read a page that said that the limit as $x$ approaches infinity of (polynomial function)/(logarithmic function) = infinity and that the limit as $x$ approaches infinity of (logarithmic ...
-1
votes
1answer
15 views

About lambert W function`s solution

I like to know solution of below equation. $$e^{-0.0042x}(1+0.0042x)=0.032$$ I use 'WolframAlpha' and get two real solution(x=-235.259 and 1256.97). General solution is ...
0
votes
2answers
18 views

Solving equation of form $x = -a/ln(bx)$

I have an equation that I am trying to solve, which can be reduced to the form $$ x = -\frac{a}{\ln(bx)}$$ where I am trying to solve for $x$. Mathematica says the solution is of the form $$x = ...
1
vote
1answer
48 views

Differential Equation involving Lambert W function

I was wondering whether there is an explicit solution to the following differential equation $$f'(x) = g'(t)\left(f(t)\left(\frac{a}{g(t)} -1 \right)-\frac{a}{g(t) \lambda}\left( 1+ ...
0
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0answers
15 views

Inverse of a Function including Lambert W

Given the function: \begin{equation} Λ_{c,d,r}(x)=rx^{c-1}\left[ 1-\frac{1-(1-c)r}{rd}\ln x \right]^{d} \end{equation} I would like to prove that its inverse function exists and is defined as: ...
3
votes
3answers
235 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
0
votes
2answers
26 views

How do I solve for x? Do I need the Lambert W function?

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ y, d, b, and x are all real, positive numbers. How do I solve for x? Do use the lambert W function and if so how is that done? Thanks! ...
0
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2answers
69 views

Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
0
votes
1answer
30 views

Problem of simplification

When trying to solve the equation $y^y = \frac{\ln^{y(1+c)}n}{n}$ , I've found the result $$y=\frac{-\ln n}{W(-\ln^{-c}n)}$$ where $c$ is a positive constant and $W$ is the Lambert function. The ...
0
votes
1answer
30 views

Any advice on how to tackle this inequality? ($x^{a}e^{x+c}\leq b$ )

How might one go about solving the inequality: $x^{a}e^{x+c}\leq b$ where $a,b,c$ are arbitrary constants ($b\geq 0$ and $a\neq0$) for $x$. My first place would be to try and get all of the ...
0
votes
1answer
18 views

Is there a compendium of equations that are solvable in terms of Lambert W?

Such a compilation would list equations in the most general form possible along with their solutions using Lambert W, and a reference to the derivation. It would also mention equations for which no ...
3
votes
1answer
41 views

Explicit Representation of $x^{x^y}=y^{y^x}$.

How do you explicitly represent $x^{x^y}=y^{y^x}$ using the Lambert $W$ function? I started using logarithms to split it up and manipulate it to a form like xe^x. I do this semi-successfully. I go ...
1
vote
1answer
15 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
3
votes
1answer
33 views

Simplify $W(xa^x)$

I'm not sure if there is any way to simplify $W(xa^x)$. It's pretty clear that $a=e$ simplifies to $x$ or $W_k(xe^x)$, but any other value of $a$, other than trivial values like $a=0,1$, don't seem ...
0
votes
2answers
44 views

Lambert W function with natural log

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ I inserted this into Wolfram Alpha and it returned: $x = y \Bbb{W}[\frac{e^\frac{d-b}{y}d}{y})]$ y, d, b, and x are all real, ...
0
votes
0answers
15 views

What $p^{ax+b} = cx +d$ means in lambert w function?

Can somebody explain what $p^{ax+b} = cx +d$ means and how to apply the lambert w function onto this equation: $(16800)(1.1)^{n-1} = (18600)+ (1200)(n-1)$
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1answer
40 views

Solution in terms of Lambert $W$ function or infinite series

I've tried to solve $x\log x = 2(x-1)(1-ax)$ for $a\ge 0$. If $a=0$, I obtained $$x \log x = 2x-2\\ \to x(\log x-2)= -2 \\ \to x\log(xe^{-2}) = -2 \\ \to xe^{-2}\log(xe^{-2}) = -2e^{-2}\\ \to x ...
2
votes
3answers
108 views

Solve for $x$ using the lambert W function $ \frac{\ln(1+bx)}{x} = a$

Question: Solve for $x$ using the lambert W function $$ \frac{\ln(1+bx)}{x} = a$$ I've got this far: $$ \frac{\ln(1+bx)}{x} = a$$ $$ \ln(1+bx) = ax $$ $$ 1+bx = e^{ax} $$ ...
-1
votes
0answers
61 views

Solve equation in terms of x; $x−b−e^x=ae^{−x}$

Solve the following in $x$. I would like in $x$, terms $b$ and $a$ are known.I think Lambert function may be solved. Please help me with your clever comments. Thank you for the help.
0
votes
2answers
31 views

How could I solve this equation: $ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1} $ for $x$?

I want to have a solution for $x$ in this equation. $$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$ Thanks !
0
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2answers
38 views

How to derive $y^{y^n}=x$ explicit form with Lambert $W$ function

I think the answer is $y = \left( \frac{n \cdot \ln(x)}{W(n \cdot \ln(x))} \right)^{\frac{1}{n}}$, seems tricky. I'm a noob!
0
votes
1answer
29 views

is there a closed algebraic solution to x(x+a)e^x=b, a,b positive reals?

I am looking at the following equation which is solvable in terms of the Lambert-W function when $a=0$ (but it is strictly positive in my case, i.e. $a>0$): $x(x+a)e^x=b$ $(a,b>0)$ more ...
0
votes
1answer
38 views

How to derive the Lambert W function series expansion?

How do you use the Lagrange inversion theorem to derive the Taylor Series expansion of W(x)? How else can you derive a series expansion?
0
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1answer
36 views

How to derive inverse of x^x to be log(x)/W(log(x))

I understand the basics of the $W$ Lambert function, but I have problems working out some problems with it. I know the answer, but I don't know how to derive it. Help would be appreciated. $y^y=x$ is ...
0
votes
2answers
58 views

How do you solve this inequality? $\frac{1}{(n+1)4^{n+1}} < .001 $

$$\frac{1}{(n+1)4^{n+1}} < .001 $$ becomes $$ 1000 < (n+1)4^{n+1}$$ Where do you go from here? Am I supposed to plug in a table of values for n? n=1: 1000 < 32 n=2: 1000 < 192 ...
1
vote
2answers
90 views

Solving $4x = e^x$ without graphing and looking for intersection

If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection?
2
votes
2answers
40 views

For what values does this method converge on the Lambert W function?

Someone from another question had noted that the following statement $$W(x)=\ln\left(\frac x{\ln\left(\frac{x}{\ln\left(\frac x{\ln(\dots)}\right)}\right)}\right)$$ Can be found from the identity ...
0
votes
2answers
60 views

Does Lambert W (Product Log) count as an explicit solution?

Say I have an equation that I can solve in $x$ as follows: $$ x = LambertW_{-1}(y)$$ Where LambertW is the product-log function. Can I say I have an explicit solution for $x$? It looks like that, ...
0
votes
1answer
46 views

Solve $x^y+y^x=a$ for $y$?

Just as I question states, I want to solve the equation for $y$, but that is proving difficult as you cannot simply just use algebraic methods. I suspect the Lambert W function might come into play.
2
votes
2answers
64 views

Finding solution with Lambert function

I have following equation to solve for $x$ $$\ln\left(1+\frac{bx}{a}\right)=\frac{4cx}{a}$$ where $a>0,b>0$ and $c>0$. In my own attempt I replaced $1+\frac{bx}{a}$ by $y$ and with this ...
2
votes
2answers
43 views

How to find solution for this equation

I have following equation$$x\ln(1+\frac{b}{x})=b$$ where $b>0$. How to find the solution for $x$. I know how to solve equation involving $x\ln(x)$ but I don't know how to solve equation where ...
2
votes
2answers
65 views

Has $e^x = ax^2$ a general solution for all $x$?

I was fiddling around with some math and stumbled upon $\exp(x) = a x^2$, finding myself unable to find a solution. Does it even have a general solution $a$ for all $x$? Some googling brought me to ...
2
votes
2answers
357 views

Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^...}}}$ does not converge, that is ...
2
votes
2answers
350 views

Convergence or divergence of infinite power towers of complex numbers $z^{z^{z^{z{…}}}}$

Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence ...
2
votes
1answer
58 views

A strange identity related to the imaginary part of the Lambert-W function

Working on a problem in QFT, i was stumbeling about some expressions containing the Lambert-$W$ function. Playing around, i discovered experimentally that the following statement seems to be true ...
4
votes
1answer
96 views

How to solve $\ln(y)=\ln(x)e^{\ln(x+1)} $ for x?

I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do $y = x^x$ // $x = e^{\ln(x)}$ $y=x^{e^{\ln(x)}}$ // $\ln$() $\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I ...
0
votes
2answers
29 views

Solving an exponential equation with x as a base and an exponent

So here's the problem: $x+3=3^x$ Obviously, graphing both sides and finding the intersection would reveal the answer, but algebraically, how can this be solved?
2
votes
1answer
57 views

Any way we can evaluate the infinite power tower where it diverges?

When you have: $$x=y^{y^{y^{y\dots}}}$$ You have: $$x=e^{-W(-\ln(y))}$$ ONLY when the power tower converges. But what about when it doesn't? Is there any way to justify ...
1
vote
1answer
85 views

Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after ...
9
votes
1answer
88 views

Sequence related to solutions of the equation $x^x=c$

A couple years ago I remember repeatedly pressing $\sqrt{1+ans}$ into my calculator to be astonished that my calculator gives me an answer approaching the golden ratio. I was astonished, and dug ...
0
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1answer
46 views

Solve for x, $2=e^{3x}-x$? [closed]

The Lambert W function should be able to help me with this but for the life of me I can't figure out how.
6
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1answer
111 views

New Elementary Function?

In the February 2000 issue of FOCUS magazine, a short article suggests that the Lambert W function could be introduced into curriculum as a new elementary function saying: "... a case can be made for ...
2
votes
1answer
43 views

Is the Lambert W function multivalued everywhere?

Is the Lambert W function multivalued everywhere except at $x=0$? It is obvious that $W(0)=0\implies 0=0e^0$ because $e^u\ne0$, therefore, it is the coefficient that determines such, and the only ...
5
votes
2answers
137 views

The general solution of $x^a = a^x$ for real $a >0$

What are the roots of $$f(x) = x^a - a^x$$ for real $a > 0$? Case 1: For $0 < a < 1$ there is 1 solution, $x=a$. Case 2: For $1\le a < e$ there are 2 solutions: $x=a$ and $[x>a]$. ...
5
votes
2answers
103 views

Can one find a closed form solution to $\ln x=\frac{1}{x}$,

Is there a closed form solution of the equation $\ln x=\frac{1}{x}$? I couldn't find a proof myself and I don't know any theorems that says when a closed form solution exists.
4
votes
2answers
93 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
8
votes
2answers
96 views

Solution to $e^{e^x}=x$ and other applications of iterated functions?

While trying to solve $e^{e^x}=x$, I ran into the simple solution $x=-W(-1)$. I found it by using the equation $$e^x=x$$Then powering both sides with a base $e$.$$e^{e^x}=e^x$$Now note that the left ...
6
votes
1answer
108 views

Solution to $xe^{e^x}$

The problem $xe^{e^x}=e$ came up another day and I wondered if it were solvable. My attempt was the following substitution,$$x=W(u)$$$$W(u)e^{e^{W(u)}}=e$$Where I used a Lambert W identity to get ...
0
votes
2answers
81 views

Solve for $t$: $ e^{-2t} + 2t = 4 $

How do we do this problem for other values of the constant, say 300 or -1000? Is there a general way to solve such questions? (Looking for a way to solve this with pen and paper.)
1
vote
1answer
28 views

Solution of $\frac{c - 1}{x - c} + \log \frac{c - x}{x - 1} = 0$ for $x$

I was looking for the maximum of the function $f(x) = \left(x - 1 \right) \log\frac{c - x}{x - 1}$ for $\{x,c\} \in\mathbb{R}^+$, $x\not=1$ (obviously) and $x \le c-1$. The normal way to find such ...