Exponential equations

[eternal kindle]

Hello. Is there any way to solve algebraically $$\frac{1}{x} - e^{-2x}=0$$?

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[eternal kindle]

Exponential equations

[wintry mortar]

Aza

[solemn lodge]

i shall attempt

e^-2x =

1/e^2x

so 1/x - 1/e^2x = 0

1/x = 1/e^2x

wait surely u can just say e^2x = x then i think

so lnx = 2x

uhh

idk what you do from there 🗿

[eternal kindle]

i ended up with

-2xe^-2x = 0

idk if i messed up

[solemn lodge]

well you’re supposed to find the value of x no?

[eternal kindle]

yes

wait, u get

$1 - xe^{-2x}=0$

[wintry mortar]

Aza

[eternal kindle]

how do u solve this

[solemn lodge]

well 1 = xe^-2x

xe^-2x = x/e^2x

1 = x/e^2x

[eternal kindle]

and?

then what?

[solemn lodge]

uhh

e^2x = x

well then u just get the same as me which doesn’t help

maybe

this may be dumb but u could say 1/1 = x/e^2x

wait nah

do u have the answer ?

[eternal kindle]

no

[solemn lodge]

hmm

if lnx = 2x

[eternal kindle]

?

[solemn lodge]

^

[eternal kindle]

where does that even come from

[solemn lodge]

read what i wrote

above that

idk if it’s correct tho

u could also say xe^2x = 1 from that but again idk how that helps

[odd steeple]

One way you could progress would be to use the Lambert-W function

[eternal kindle]

oh true

i need to multiply and divive by 2

[odd steeple]

Essentially, yeah

[eternal kindle]

well, okey

[solemn lodge]

bruh deadass

[odd steeple]

Because Lambert-W is defined as the inverse of f(x) = xe^x

[solemn lodge]

isn’t this like an 11th/12th grade question

[odd steeple]

Aside from Lambert-W, I’m not sure if you can get a algebraic closed-form expression with equations of the form xe^x = c, where c is an integer

[eternal kindle]

newtons method i guess

but thats not algebraically

so W is the way

[odd steeple]

Possibly, I’m unsure though

[odd steeple]

Yeah, that’s numerical

[eternal kindle]

so it will be

[solemn lodge]

i mean im not familiar with lambert W function but i don’t see any other way to solve it

[odd steeple]

Correct me if I am mistaken, but I believe it has been proven that you can’t write a closed form expression for this equation

[eternal kindle]

$W(-2xe^{-2x}) = W(-1/2)$?

[wintry mortar]

Aza

[eternal kindle]

then $-2x = W(-1/2) \implies x = W(-1/2)/-2$

[wintry mortar]

Aza

[solemn lodge]

does W have a value?

[odd steeple]

I don’t think so,

[odd steeple]

After we reach this

[eternal kindle]

does W give u x back?

or whatever?

[odd steeple]

Then 1 = xe^(-2x)
Then -2 = (-2x)e^(-2x)

[eternal kindle]

and W gives u back 2x? or x?

[odd steeple]

It would give us back -2x (I didn’t see the “-“ sign)

[odd steeple]

So it would follow that -2x = W(-2)

[eternal kindle]

ah okey

okey, my bad with algebra

i see

[odd steeple]

Essentially, W(xe^x) = x

[odd steeple]

That’s fine!

[eternal kindle]

thanks tho, i forgot about W

[odd steeple]

No problem!

[eternal kindle]

and can u manually compute W? or how is it defined?

[odd steeple]

It’s defined as the inverse of f(x) = xe^x

So W(xe^x) = x

However, I believe it has a “nice” taylor series or approximation method.

[eternal kindle]

yeah but that doesnt help, like, how do i know what is W(-2)?

[signal meteor]

W is a math function just like any other

you can include it in the things you allow yourself to use to define things

like you could just as well ask "can you manually compute sin(x)"

[eternal kindle]

again, W(-2) has a value, how do i compute it?

[signal meteor]

yeah you can with numerical

[odd steeple]

Yeah, sort of like that

[signal meteor]

but you can't really express it in terms of other things like operations, square root, etc

it is only itself

[eternal kindle]

for the third time xd How do i compute it?

[odd steeple]

Best you could do would probably be to use a numerical approximation method

Something like this, or other approximation methods.

[eternal kindle]

ah

okey

ty

[odd steeple]

No problem!

Do note though, that you won’t obtain the “exact” value, most likely an “approximate”. @eternal kindle

[eternal kindle]

well, same way u compute any other irrational number