All Questions
13
questions
2
votes
2
answers
133
views
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
4
votes
6
answers
465
views
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
votes
4
answers
254
views
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
-1
votes
1
answer
199
views
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.
$$ \ln\left(\frac{GC}{a}\right) = hP + \ln(P) $$
- 1
-1
votes
1
answer
79
views
All real and closed-form roots of $\log_2x=\frac {2^{x-1}}{x}$
What are the closed-form roots of
$$\log_2x=\frac {2^{x-1}}{x}$$
?
My attempts:
Closed-form means, I assume that the Lambert W function can work.
I know that, at least $x>0$.
Wolfram Alpha gives ...
- 1,659
5
votes
5
answers
228
views
Is there an algebraic solution to $\log_{\sqrt2}{\left(x\right)} = (\sqrt2)^x$?
I’m trying to solve
$$\log_{\sqrt2}{\left(x\right)} = (\sqrt2)^x$$
My next step is
$$\ln{x}= (\sqrt2)^x\ \cdot\ \ln\sqrt2$$
EDIT:
I’m only up to high school math.
- 123
1
vote
0
answers
285
views
curve equation from $\frac{1}{2^x}-\frac{1}{3^x}=\frac{1}{2^y}-\frac{1}{3^y}$
Is there a way to find the equation of the curved part for this:
$$\frac{1}{2^x}-\frac{1}{3^x}=\frac{1}{2^y}-\frac{1}{3^y}$$ for $x$?
See WolframAlpha for the plot.
The curve is crossing the line $y=x$...
- 308
0
votes
1
answer
157
views
Solving Equation involving Lambert W function
I have the following equation that depends on the values $a, q, x$ where $a,q>2$ are positive non-zero real numbers.
$$
y = \log(x\log(2) + a) - \left(2-\frac{1}{2^x}\right) q
$$
Now I was able to ...
- 3,105
0
votes
1
answer
76
views
How to find $x$ for $-\frac{1}{\sqrt{2}x^{\frac{3}{2}}}=-\frac{1}{4}e^{-\frac{x}{4}}\left(A-B\right)$
The title pretty much explains it; I've had trouble with this because when taking the logarithm of both sides $x$ can never be isolated. I have been looking into the Lambert W function, but I've never ...
user809100
2
votes
2
answers
196
views
What is the function $y = (1+1/x)^x$ solved for $x$?
I came across this function in algebra ($e$ being its limit as $x$ goes to infinity) while studying compounded interest. Since this function is a little modified from the real interest formula $y=(1+1/...
- 164
1
vote
2
answers
127
views
How does one solve this kind of equation: $3^x=x+3$
How does one solve this kind of equation: $$3^x=x+3$$
I tried playing around with logs but it didn't get me anywhere.
I plotted the two functions $f(x)=3^x$ and $g(x)=x+3$ on a graph to estimate the ...
- 2,731
2
votes
0
answers
74
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 ...
- 566
4
votes
2
answers
143
views
Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$?
Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$? My suspicion after a fruitless hour of manipulation is that it is not.
- 183