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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4
votes
1answer
67 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
27 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?
0
votes
0answers
26 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 ...
0
votes
1answer
70 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
83 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
votes
1answer
41 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
votes
1answer
99 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
36 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
125 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 $[x>a]$. ...
6
votes
2answers
92 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
89 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
90 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
99 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
75 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.)
0
votes
0answers
45 views

Solving for $x$ in $(y-x)\ln\frac{x}{y} = a$

I have the expression $$(y-x)\ln\frac{x}{y} = a,$$ and I want to express $x$ in terms on $y$ and $a$. I know that in this kind of problem, the Lambert function $W$ is likely to show-up, but that ...
1
vote
1answer
27 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 ...
0
votes
1answer
43 views

Is there a formula to find the exact value of inverse factorials?

$x!=y, \space x ∈ ℝ$ Is there a formula to find the exact value of $x$ in this case, assuming that we know the value of $y$? I could do $L(x)/W(\frac{L(x)}{e}) + \frac{1}{2}$ where $W$ is the ...
1
vote
3answers
49 views

“Good approximation” for the inverse function of $y = x\log_2 x, \hspace{2mm}x>1 $?

I encountered to solve $x$ from $y$ in the equation $y = x\log_2 x ,\hspace{2mm} x>1$, which is known to have no closed form for its inverse function ...
1
vote
2answers
41 views

Deriving a differential equation for the Lambert W function.

I would like to know if I am correct in the following: Let $f(x) = e^{x}x$, and $g(x) = w(x)$, where $w(x)$ is the Lambert W function. By the rule that inverse function integral relation, which ...
1
vote
1answer
30 views

Two exponential terms equation solution

Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using lambert W function. But can I get a general solution of this equation? $$A_1 ...
1
vote
1answer
30 views

An approximation for the Lambert W-function

Proposition Let $f(x) = k^{x}x$, where the values of both $f(x)$ and $k$ are known. Let $x_{0} = f(x)$, and: $$x_{n + 1} = \frac{1}{2}\log_{k}{\left(\frac{k^{x_{n}}x_{0}}{x_{n}}\right)}$$ ...
0
votes
0answers
36 views

How to solve $n\ln^{2}(\ln 2^{n}) = g(k)$ for $n$?

I've been trying to find the inverse of an asymptotic function for personal research, and I've gotten it down to: $$n\ln^{2}(\ln 2^{n}) = \exp(\frac{9}{64}\ln^{3}(2^{k})))$$ where $\ln n$ is the ...
0
votes
0answers
17 views

Finding the intersection of two parametric equations? || How does one solve $x^x = n$? [duplicate]

Specifically: $c_1: x = t^t, y=t$ and $c_2: x= 81, y=t$ When trying to solve it, I'm coming up with: $y^y = 81$, and $y = y$ which is basically what I started when trying to solve $x^x = 81.$ ...
0
votes
3answers
85 views

How do you solve this equations where the unknown is to the power of the unknown?

How do you solve equations like: $$x^x=7$$ I've been thinking about this but couldn't find any answer. (I'm not looking for graphical solutions, only pure algebra) Thanks!
0
votes
2answers
61 views

If $y=ax^be^{-cx}$ then $x=g(y)$, find $g$

I have this function $$y=0.384394\cdot x^{0.341429}\cdot e^{-0.004749 x}$$ Based on this function I would like to know how I can I get $x=g(y)$.
0
votes
0answers
23 views

Does LambertW have complex branches?

The $W(x)$ function (LambertW) has two branches for $x \in [-e^{-1}, \infty)$. This is a very interesting property. Suppose we have the following: $$ -e^{-1} = -1 \cdot \exp(-1) \ne (-1 + 2in\pi) ...
0
votes
1answer
46 views

Inverse of a function

How can we find the inverse of $f(x) = 3x + e^{2x} $? I am not able to separate the $x$ even taking the logarithm on both sides.
0
votes
0answers
17 views

An algebraic equation, summing exponential and other terms

I am looking for a solution of an algebraic equation which seems simple... Here it is, for a,b,c,d four constant real numbers, and x the unknown which is also real: $$ x + e^{ax} + be^{cx} + d = 0 $$ ...
1
vote
0answers
49 views

How can $x = \frac{b}{\ln(x + a)}$ be solved for $x$?

I've solved $x = \ln(x + a)$ by $x = -W(-e^{-a}) - a$, so I suspect that this will also involve the Lambert W function. However, I've been unable to make any progress due to the $x + a$, but without ...
2
votes
1answer
44 views

Doubt in raising a power to a complex number [duplicate]

What's the value of $$i^{i^{i^{...}}}$$? I tried to take log on both sides. $x=i^x$ $\implies \log x=x \log i$ After this how can I solve this... I am sorry, that I don't know the methods you ...
5
votes
3answers
281 views

Using Lambert's W Function to solve this equation

I'm attempting to solve the following equation (eventually with Lambert's W Function having checked the solution on Wolfram Alpha): $$100n^2 = 2^n$$ I got as far as follows but I am unsure how to ...
1
vote
1answer
53 views

Is the following solvable for x?

I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base: $[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt ...
2
votes
1answer
49 views

simplification of $W(x\cdot e^{a+x})$

Is it possible to simplify $W(x\cdot e^{a+x})$? Because $W(x\cdot e^{x})=x$ So I was wondering if it was possible to simplify this expression.
0
votes
0answers
28 views

solve implicit equation with lambertw, exponentials, logarithms and first order polynom

I have a very complicated problem to solve. I am almost sure it's impossible to solve but maybe one of you guys has a miracle solution for me. I am modelling the behaviour of a photovoltaic cell and ...
1
vote
2answers
100 views

Find the roots of a function with logarithms (possibly using lambert W function)

I am wondering if anyone can help me find an analytical solution to the roots of the following function: $$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$ $a,b ...
2
votes
1answer
155 views

What is the solution of $a^b=a+b$ in terms of $a$?

Let $a, b$ be real numbers. Solve $$a^b=a+b$$ for $a$. If there isn't a solution with $a, b$ real, maybe $a, b$ should be complex. But no matter how hard I try, this is proving to be very ...
0
votes
0answers
13 views

Compute one branch of Lambert W function from the other

Assume that I have one real solution to $W(x) = x \cdot \exp(-x) = y$, so I know $(x_1, y)$ such that $x_1 \cdot \exp(-x_1) = y$. Is there any easy way to find the second solution $x_2 \neq x_1$ such ...
3
votes
1answer
72 views

How could I solve $x^{t-1}e^{-x} = a$ for $x$?

Consider this equation: $$x^{t-1}e^{-x} = a$$ I am aware that this is what you integrate from $0$ to $\infty$ in respect to $x$ to get the Gamma Function, but I do not want to worry about it here. I ...
2
votes
1answer
74 views

Solving $x - a \log(x)=b$

Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form?
7
votes
1answer
94 views

Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?

According to this Wikipedia page, the Lambert W relation cannot be expressed in terms of elementary functions. However, it does not explain why this is the case. An elementary function is "a ...
0
votes
1answer
65 views

Solving an exponential function

I have the below exponential function which I wish to solve it for $b$. Other than resorting to the Lambert W function, is there alternative way of representing the solution? $$ \frac{(1+a)(1-b)}{ab ...
1
vote
2answers
57 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
13
votes
2answers
280 views

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
2
votes
1answer
58 views

Solve $\exp(x)(5-x)=5$ by hand

Is there a way to solve this equation by hand? $\exp(x)(5-x)=5$ Solutions: $x_1=0$ $x_2= 4.96511$
9
votes
0answers
156 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
4
votes
1answer
60 views

Methods for Finding Exact Solution For $e^{2x}+p(2x)$

I know there are ways using the Lambert W function, and have had answers to simpler examples, for example $$e^{2x}+1+2x=0\Rightarrow e^{2x}=-2x-1$$ has the solution ...
7
votes
2answers
607 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was ...
3
votes
0answers
72 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
0
votes
0answers
29 views

Where did I go wrong in this simplification involving the Lambert W Function?

I have been working on this problem for about a day and thought I finally found a way to simplify it, unfortunately when I plug my simplification back into maxima I get that they are not equal by what ...
0
votes
1answer
29 views

Big Theta of this modification of the secondary branch of the Lambert W function

I am looking to find the big-$\Theta$ of $-W_{-1}(-\frac{a}{n})$ in terms of elementary functions where $a$ is a constant. Looking around and I find that this should be $O(\log(n))$ and with maxima I ...