# Tagged Questions

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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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 $[x>a]$. ...
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### 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.
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### 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 $a, b, c,$ ...
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### 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 ...
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### 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 ...
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### 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.)
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### 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 ...
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### 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 ...
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### 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 ...
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### “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 ...
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### 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 ...
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### 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.
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### 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: $$x + e^{ax} + be^{cx} + d = 0$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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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? $$\frac{(1+a)(1-b)}{ab ... 2answers 57 views ### Finding the inverse of the function f(k, x) = k^{x}x. Recently, I have been looking at the function f(x) = e^{x}x, where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ... 2answers 280 views ### Proof of strictly increasing nature of y(x)=x^{x^{x^{\ldots}}} on [1,e^{\frac{1}{e}})? The title is fairly self explanatory: I have been trying to rigorously prove that y(x)=x^{x^{x^{\ldots}}} is a strictly increasing function over the interval [1,e^{\frac{1}{e}}) for a while now, ... 1answer 58 views ### Solve \exp(x)(5-x)=5 by hand Is there a way to solve this equation by hand? \exp(x)(5-x)=5 Solutions: x_1=0 x_2= 4.96511 0answers 156 views ### Could this approximation be made simpler ? Solve n!=a^n 10^k I need to find the smallest value of n such that$$\frac{a^n}{n!}\leq 10^{-k}$$in which a and k are given (these can be large numbers). I set the problem as : solve for n the equation ... 1answer 60 views ### Methods for Finding Exact Solution For e^{2x}+p(2x) I know there are ways using the Lambert W function, and have had answers to simpler examples, for example$$e^{2x}+1+2x=0\Rightarrow e^{2x}=-2x-1$$has the solution ... 2answers 607 views ### Is there ANY possible way to solve this equation? So I came up with this equation and it just seems like I can't solve it AT ALL for 'a'$$a*b^a = c$$EDIT: By the way, I'm only taking b^a, not both b and a, just in case anyone was ... 0answers 72 views ### Mapping exponential functions in polar coordinates I tried mapping power functions onto the polar plane (i.e. converting x,y into r and \theta). I was successful with power functions representing y=ax^n by$$r=\sqrt[n-1]{\frac ...
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 ...