/sci/ - Science & Math » Thread #13958673

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x + lnx = 2

Anonymous Mon 13 Dec 16:52:05 2021 No.13958673 View Reply Original Report

Quoted By: >>13958691 >>13958705 >>13958765 >>13958792 >>13958899 >>13958934 >>13959062 >>13959212 >>13959255 >>13959352 >>13959449 >>13959764 >>13959825 >>13959992 >>13960835

is this actually impossible to solve algebraically, wtf bros?

Anonymous

Anonymous Mon 13 Dec 2021 16:56:50 No.13958689 Report

Quoted By: >>13958690

x+ e^2 = x

No real solution.

Anonymous

Anonymous Mon 13 Dec 2021 16:57:40 No.13958690 Report

Quoted By:

>>13958689
learn math

Anonymous

Anonymous Mon 13 Dec 2021 16:57:44 No.13958691 Report

Quoted By: >>13958699 >>13959212 >>13959828

>>13958673
>Test x = 1 see it's too small
>Test x = 2 see it's too large
>By continuity and monotony there is a unique solution in the interval ]1,2[
>Now test x = 1.5. See that it's again too small so the solution is in the interval ]1.5,2[
>Now test x = 1.75 and so on...
With each step the interval where the solution lies halves in size.

Anonymous

Anonymous Mon 13 Dec 2021 17:01:16 No.13958699 Report

Quoted By:

>>13958691
This is how I brute force exam questions I can't solve just lolm

Anonymous

Anonymous Mon 13 Dec 2021 17:02:13 No.13958705 Report

Quoted By: >>13959212 >>13959608

>>13958673
"exact" solutions are useless anyway
for all practical problems an approximation is sufficient

Anonymous

Anonymous Mon 13 Dec 2021 17:20:50 No.13958765 Report

Quoted By:

>>13958673
i think its impossible
T. failure of a calc 3 student

Anonymous

Anonymous Mon 13 Dec 2021 17:29:31 No.13958792 Report

Quoted By: >>13958860 >>13959212 >>13962198

>>13958673
It is impossible to solve with elementary functions bc lambert W. However, you can get a power series for the answer.

Anonymous

Anonymous Mon 13 Dec 2021 17:45:12 No.13958860 Report

Quoted By: >>13959212 >>13962198

>>13958792
Yeah Lambert W makes this trivial

Anonymous

Anonymous Mon 13 Dec 2021 17:51:38 No.13958899 Report

Quoted By: >>13958984 >>13959010 >>13959051 >>13959054

>>13958673
One solution would be x = 1.

Anonymous

Anonymous Mon 13 Dec 2021 18:00:23 No.13958934 Report

Quoted By:

>>13958673
Solve with Newton method

Anonymous

Anonymous Mon 13 Dec 2021 18:09:32 No.13958984 Report

Quoted By: >>13959051

>>13958899
true, didn't think about that

Anonymous

Anonymous Mon 13 Dec 2021 18:15:42 No.13959010 Report

Quoted By: >>13959021

>>13958899
no

Anonymous

Anonymous Mon 13 Dec 2021 18:20:30 No.13959021 Report

Quoted By: >>13959034 >>13959051 >>13959064

>>13959010
Nigger how retarded are you?

Anonymous

Anonymous Mon 13 Dec 2021 18:24:29 No.13959034 Report

Quoted By:

>>13959021
Shut up you retard faggot grifter kill yourself and your mother you nigger.

Anonymous

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Anonymous Mon 13 Dec 2021 18:28:52 No.13959051 Report

Quoted By:

>>13958899
>>13958984
>>13959021

Anonymous

Anonymous Mon 13 Dec 2021 18:29:22 No.13959054 Report

Quoted By:

>>13958899
That is incorrect dumb faggot

Anonymous

Anonymous Mon 13 Dec 2021 18:31:07 No.13959062 Report

Quoted By: >>13959212

>>13958673
There is one solution, which i just verified from a graphing calculator but i dont thing you can sove this equation just by simple algebraic manipulations.

Anonymous

Anonymous Mon 13 Dec 2021 18:31:54 No.13959064 Report

Quoted By: >>13959068

>>13959021
Ln1 equals zero genius

Anonymous

Anonymous Mon 13 Dec 2021 18:33:08 No.13959068 Report

Quoted By: >>13959075

>>13959064
Yes, so we're 1 short of adding up to 2.

Anonymous

Anonymous Mon 13 Dec 2021 18:35:01 No.13959075 Report

Quoted By:

>>13959068
Wrong @

Anonymous

Anonymous Mon 13 Dec 2021 18:35:39 No.13959077 Report

Quoted By: >>13959255

x = W(2)

Anonymous

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Anonymous Mon 13 Dec 2021 19:10:54 No.13959212 Report

Quoted By: >>13959550 >>13962231

>>13958673
Let's try substitution:
$x+\ln x=2$
$x=2-\ln(x)$
$x=\ln(2-\ln(2-\ln(2-\ln(2-\ln(2-\dots)))))$

Weird...Ok, let's try that again:
$x+\ln x=2$
$y=\ln(x) \iff e^y=x$
$e^y+y=2$
$e^y=2-y$
$y=\ln(2-y)$
$y=\ln(2-\ln(2-\ln(2-\ln(2-\ln(2-\dots)))))$
What? $x=y$ ? so $e^x=x$ ? or equivalently $x=\ln x$ ?

There can't be a solution, right? someone help, my schooling sucked so I had to self-learn math.
>>13958691
Seems inelegant.
>>13958705
Not very mathy of you, bro.
>>13958792
>>13958860
What the fuck are you mathfags on about?
>>13959062
Is this what people mean by non-algebraical number? how can algebra not be enough? I don't get it...

Anonymous

Anonymous Mon 13 Dec 2021 19:22:07 No.13959255 Report

Quoted By: >>13959331

>>13959077
wrong.

W(2) + ln (W(2)) = 0.693.... = ln 2

>>13958673
Lets use x=4 for example: $4+ln(4) = 5.386294....$

So it means, there exists a solution. If you ask for let's say $x+ln (x) =5.386294....$ there has to be a way to get the solution "4".

So there has to be a solution for $x+ln(x) = 2$

Anonymous

Anonymous Mon 13 Dec 2021 19:25:17 No.13959264 Report

Quoted By: >>13959566

ALMOST ALL mathematical promlems have NO CLEAN PEN AND PAPER SOLUTION

El Arcón

El Arcón Mon 13 Dec 2021 19:26:55 No.13959269 Report

Quoted By:

The logarithm of x is an infinite series in x so that's not really very unbelievable that the equation would be intractible.

Anonymous

Anonymous Mon 13 Dec 2021 19:33:11 No.13959294 Report

Quoted By: >>13959344

The solution is about x=1.558

Anonymous

Anonymous Mon 13 Dec 2021 19:38:39 No.13959314 Report

Quoted By:

The solution is about tree fiddy.

Anonymous

Anonymous Mon 13 Dec 2021 19:46:46 No.13959331 Report

Quoted By:

>>13959255
oh right i forgot an operation. it's W(exp(2))

Anonymous

Anonymous Mon 13 Dec 2021 19:52:04 No.13959344 Report

Quoted By:

>>13959294
so half of pi. nice.

Anonymous

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Anonymous Mon 13 Dec 2021 19:54:11 No.13959352 Report

Quoted By: >>13959441

>>13958673
solved. now kys.

Anonymous

Anonymous Mon 13 Dec 2021 20:18:46 No.13959441 Report

Quoted By: >>13959599

>>13959352
Literally shows the value is not 2 on your plot
>Kys

Anonymous

Anonymous Mon 13 Dec 2021 20:20:03 No.13959449 Report

Quoted By:

>>13958673
exp(x+ln(x))=exp(2)
exp(x)*exp(ln(x))=exp(2)
x*exp(x)=exp(2)
x=W(exp(2))

Anonymous

Anonymous Mon 13 Dec 2021 20:38:55 No.13959542 Report

Quoted By: >>13962346

Seriously, someone explain ----> 13959212.

Anonymous

Anonymous Mon 13 Dec 2021 20:40:57 No.13959550 Report

Quoted By: >>13959631 >>13959656 >>13959857 >>13962212

>>13959212
x = lnx = z
2z = 2
z = 1
x=1 = 0
qed

Anonymous

Anonymous Mon 13 Dec 2021 20:48:51 No.13959566 Report

Quoted By: >>13959620 >>13962203

>>13959264
That's why numerics chads always win

Anonymous

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Anonymous Mon 13 Dec 2021 20:58:53 No.13959599 Report

Quoted By:

>>13959441
kys

Anonymous

Anonymous Mon 13 Dec 2021 21:00:39 No.13959608 Report

Quoted By: >>13959610 >>13959765

>>13958705
>what is chaos

Anonymous

Anonymous Mon 13 Dec 2021 21:01:28 No.13959610 Report

Quoted By:

>>13959608
A ladder

Anonymous

Anonymous Mon 13 Dec 2021 21:03:53 No.13959620 Report

Quoted By: >>13959657

>>13959566
>Fancy guessing

Anonymous

Anonymous Mon 13 Dec 2021 21:07:11 No.13959630 Report

Quoted By:

Simply use the same method that you would use to solve x + ln x = 1.

Anonymous

Anonymous Mon 13 Dec 2021 21:07:26 No.13959631 Report

Quoted By: >>13959732

>>13959550
Ok, but where is the fallacy? I assume it's the last step but how?

Anonymous

Anonymous Mon 13 Dec 2021 21:15:10 No.13959656 Report

Quoted By: >>13959694

>>13959550
>2z = 2
what?

Anonymous

Anonymous Mon 13 Dec 2021 21:15:21 No.13959657 Report

Quoted By:

>>13959620
It's not guessing, we always know how big the error is, and we keep iterating until it's sufficiently small for our purpose.

Anonymous

Anonymous Mon 13 Dec 2021 21:27:46 No.13959694 Report

Quoted By:

>>13959656
For some reason he changed my y variable to z.
By definition y = ln x.
By my (fallacious) reasoning I got x = y and therefor x = ln x.
Since x = ln x, x + ln x = 2 is the same as saying y + y = 2 or 2y = 2.
Furthermore, since y = 1 and x=y and x = ln x then 1 = ln 1 so 1 = 0.
I don't understand my mistake but clearly the conclusion is ridiculous.

Anonymous

Anonymous Mon 13 Dec 2021 21:36:57 No.13959725 Report

Quoted By:

x cannot equal lnx

e^1=e
e^0=1
ln1 = 0
ln0 is undefined
Look at the graph. That graph never touches the line where y = x and so the input could never equal the output.

Anonymous

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Anonymous Mon 13 Dec 2021 21:40:07 No.13959732 Report

Quoted By:

>>13959631

forgot pic

That's a simple trick you can use anytime someone says x = some function of x.
If the graph of the function doesn't intersect y=x, or pass through the origin, then there is no solution

Anonymous

Anonymous Mon 13 Dec 2021 21:48:54 No.13959764 Report

Quoted By:

>>13958673
why is it surprising
try x - x =2
same deal.

Anonymous

Anonymous Mon 13 Dec 2021 21:49:55 No.13959765 Report

Quoted By:

>>13959608
locally linear

Anonymous

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Anonymous Mon 13 Dec 2021 22:11:24 No.13959825 Report

Quoted By: >>13959845 >>13959860

>>13958673
0.159 and pi

Anonymous

Anonymous Mon 13 Dec 2021 22:12:32 No.13959828 Report

Quoted By:

>>13958691
based binary searcher

Anonymous

Anonymous Mon 13 Dec 2021 22:17:06 No.13959845 Report

Quoted By:

>>13959825
you read the question wrong

Anonymous

Anonymous Mon 13 Dec 2021 22:21:13 No.13959857 Report

Quoted By:

>>13959550
you just showed that there are no solutions to x+lnx=2 with x=lnx which is correct
theres only one solution and it has x != lnx

Anonymous

Anonymous Mon 13 Dec 2021 22:22:17 No.13959860 Report

Quoted By: >>13962212

>>13959825
based wrong answer giver

Anonymous

Anonymous Mon 13 Dec 2021 23:04:37 No.13959992 Report

Quoted By:

>>13958673
You can do a change of variables $z = x - 1$ to get the equation $z + \ln(1+z) = 1$ . From here, you can use a peturbation series on $z + \ln(1+z) = \varepsilon$ where $z = \sum_{n=0} z_n \varepsilon^n$ to get a series form of the solution in terms of z, and then simply add 1 to get a series form for x. The exact solution can't be expressed in elementary functions, so a series solution is the best you will get. I got for the first few terms, $z_0 = 0, z_1 = \frac{1}{2}, z_2 = \frac{1}{16}, z_3 = \frac{1}{192}$ so as you can see this converges pretty quickly.

Anonymous

Anonymous Tue 14 Dec 2021 03:18:27 No.13960835 Report

Quoted By: >>13961655

>>13958673
in terms of lambert w, it would be:
$x+\ln(x)=2$
$xe^x = e^2$
$W(xe^x) = W(e^2)$
$x = W(e^2)$

right, anons?

Anonymous

Anonymous Tue 14 Dec 2021 11:16:22 No.13961655 Report

Quoted By: >>13961762 >>13962172 >>13962231

>>13960835
What is this shit? answer my question here ----> 13959212.

Anonymous

Anonymous Tue 14 Dec 2021 12:00:13 No.13961762 Report

Quoted By:

>>13961655
That ist the answer

Anonymous

Anonymous Tue 14 Dec 2021 14:46:02 No.13962172 Report

Quoted By:

>>13961655
https://letmegooglethat.com/?q=Lambert+W+function

Anonymous

Anonymous Tue 14 Dec 2021 14:56:13 No.13962189 Report

Quoted By:

>>13961776
She says you can't solve it. Is she right?

Anonymous

Anonymous Tue 14 Dec 2021 15:01:21 No.13962198 Report

Quoted By:

>>13958860
>>13958792
Shieeet I remember I needed to use this function in my MSc thesis because a stochastic control solution was a differential equation that fell into these cases.

Anonymous

Anonymous Tue 14 Dec 2021 15:03:22 No.13962203 Report

Quoted By: >>13962299

>>13959566
to be fair, perturbation chads do much better and avoid the shit numerical libraries full of unknown bugs and floating point shit

the only issue is that increasing a term in the perturbative expansion is hard as fuck

Anonymous

Anonymous Tue 14 Dec 2021 15:06:32 No.13962212 Report

Quoted By:

>>13959860

I think anon was answering >>13959550 not op. But It doesn't matter; it is still relevant. By shifting either x by 2 or lnx by -2 it might cross y=x giving the one x and y that also solves x = lnx.

Anonymous

Anonymous Tue 14 Dec 2021 15:13:12 No.13962231 Report

Quoted By: >>13962737

>>13959212
>>13961655
if a = ln(2-ln(2-...)), then x = 2 - a, not x = a as you've incorrectly written.

Anonymous

Anonymous Tue 14 Dec 2021 15:39:49 No.13962299 Report

Quoted By:

>>13962203
>the only issue is that increasing a term in the perturbative expansion is hard as fuck
Seriously lol, this shit is easy to do up to third order and then goes bit haywire 4th order. Can't imagine going any higher. Luckily it converges very quickly.

Anonymous

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Anonymous Tue 14 Dec 2021 15:51:26 No.13962346 Report

Quoted By: >>13962471

>>13959542

Anonymous

Anonymous Tue 14 Dec 2021 16:37:14 No.13962471 Report

Quoted By: >>13962585

>>13962346
Go back where, faggot?

Anonymous

Anonymous Tue 14 Dec 2021 17:10:58 No.13962585 Report

Quoted By:

>>13962471
to old /b/ I guess? where else is there?

Anonymous

Anonymous Tue 14 Dec 2021 17:44:24 No.13962737 Report

Quoted By:

>>13962231
Oh, I see:
$x=2-\ln(2-\ln(2-\ln(2-\ln(2-\ln(2-\dots)))))$
Therefor, $x=2-y$ or $x=2-\ln(x)$ so there's no contradiction but we've also seemingly learned nothing, huh? bummer.

Thanks bro.

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