All Questions
Tagged with logarithms lambert-w
113
questions
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Bounding the solution of a logarithmic equation
Given a small number $\varepsilon >0$ and a constant $1/3\le \alpha < 1 $, I am looking for the smallest possible number $x^*$ such that for all real $x\ge \max\{x^*,3\}$, we have
$$\frac{x}{(\...
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2
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256
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Solve $x^x-5x+6=0$ using Lambert W function.
How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function?
EDIT:
I solved equations $2^x - 5x + 6 = 0$ and $3^x - 4x - 15 = 0$ using Lambert W function, but not able to solve $x^x - 5x + 6 = 0$ ...
1
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2
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How to solve $\ln(x) = 3\left(1-\frac{1}{x}\right)$?
I have been working for this problem for a while:
$$\ln(x) = 3\left(1-\frac{1}{x}\right)$$
and by graphing, plugging and chugging values and rigorously doing the math, I can clearly see that one of ...
- 69
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2
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Need your help in solving $\log_\sqrt[34]{2} x = 4x^4-3x^3-2x^2+x$
I have been playing with graphs until made a nice equation
$$\log_\sqrt[34]{2} x = 4x^4-3x^3-2x^2+x$$
The real answers are 1 and 2. But how to solve it? And is it possible to stretch it to complex ...
- 39
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How do you find solutions to the equation $\frac{x}{y^\frac{x}{y}} = 1$, which involves the Lambert W function?
How do you solve for $x$ in the following equation:
$$
\frac{x}{y^\frac{x}{y}} = 1
$$
By graphing it, I know there are two real solutions. I tried doing it in the following way:
$$
x = y^\frac{x}{y} \\...
5
votes
2
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186
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Equations similar to Lambert-$W$ with quadratic exponents
I've seen solutions saying that an equation in the format:$$ \ln(x) - \frac{bx}{a} = - \frac{bc}{a} $$
can be solved using the Lambert W function and I am comfortable doing so. My equation however is ...
2
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2
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133
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Does the second positive solution (besides $x = 1$) of the equation $e^{x^2-1}=x^3-x\ln x$ have a closed form?
Does the equation $$e^{x^2-1}=x^3-x\ln x$$ have a closed form solution ?
The given equation has $2$ positive real roots. Graphically
It is not hard to see that $x=1$ is a rational solution. The ...
- 644
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3
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Finding a real closed-form solution to a tricky transcendental equation
One of my friends posed the following question with a prize of free bubble tea for anyone who could find a closed-form solution to the problem (not necessarily in terms of elementary functions):
What ...
- 75
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6
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How can we find Lambert W solution to $\dfrac {x\ln x}{\ln x+1}=\dfrac{e}{2}$?
Find all real solutions: $$\frac {x\ln x}{\ln x+1}=\frac{e}{2}$$
Cross multiplication gives $$2x\ln x=\ln (x^e)+e$$ I didn't see any useful thing here. I tried solving this equation in WA. The ...
- 644
2
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4
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254
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How to solve $x + 3^{x} = 4$ using Lambert W Function.
As stated in the title I am trying to solve the equation $$x + 3^{x} = 4$$ using Lambert W Function and which led me to the result $$x = 4 - \frac{W(3^{4} \ln{3})}{\ln{3}}$$ and driven by the belief ...
- 193
2
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3
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122
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Showing that $\ln(x)^2+2(x-1)\ln x-3x+1=0$ has only $2$ real solutions
Is there any elementary way to show that $\ln (x)^2+2(x-1)\ln x-3x+1=0$ has $2$ real solutions on $(0,\infty)$?
I did it by this way.
Let $f(x)=(\ln x)^2+2(x-1)(\ln x)-3x+1$.
signs of $f(\frac{1}{2})$...
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3
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161
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How to rewrite $e^{-x} = -\ln(x)$ such that the Lambert W function can be applied?
Original problem:
Find the real solution of $$e^{-x} = -\ln(x)$$
The first thing I did was divide both sides by $e^{-x}$.
With a few more steps I arrived at
$$e^x\ln(x) = -1$$
This was great, but ...
- 13
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Closed form for solution of $\ln(x)\ln(x+1)=1 $ [closed]
Is it possible to find a close form for the solution of this equation (maybe with the use of Lambert W function)?
$$ \ln(x)\ln(x+1)=1 $$
- 853
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1
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127
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Inverting series with logs and W
You've all heard it: what does a drowning analytic number theorist say? Log log log log....
I very frequently deal with the sorts of functions that one comes across and want to invert them. Generally ...
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1
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How to solve for $x$ from $x + \ln(x) = \ln(c)$?
How do I solve this equation for P? For everything I've tried, P ends up trapped in an exponent or another natural log.
$$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) $$
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