I heard about the $W$ Lambert function yet what I can see on Wikipedia deserve to only resolve the equations of form $x imes e^x=lambda$. How deserve to I do?

Online solvers told me the solution was $x=e^-fracW(2)2$ but I don"t understand why...

You are watching: What is e^x times e^x


When attempting to resolve equations making use of the Lambert $W$ feature, one typically has to exploit the fact that different manipulations affect the "coefficient" in front of $e$ and also the exponent above $e$ differently.

In this case, you have actually $x^2$ in the exponent, so you desire that in the coreliable as well. We deserve to attempt squaring our equation and also check out where that gets us:$$x^2e^2x^2 = 1$$OK. This squared the coeffective yet doubled the exponent. So now we have actually $x^2$ both places. However, the exponent is $2x^2$ rather of $x^2$. So, we multiply our equaiton by $2$ to get$$2x^2e^2x^2 = 2$$Now we deserve to use the Lambert $W$ function to get$$2x^2 = W(2)\x = sqrtW(2)/2$$


Firstly, you desire to usage the fact that $x = e^ln x $, therefore the expression becomes$$x e^(e^lnx)^2 = 1, $$combining it over a single exponential$$Rightarrowhead e^lnx + e^2 ln x = 1.$$Taking the herbal logarithm, we then obtain $$ln x + e^2 ln x = 0 Rightarrowhead e^2 ln x = -ln x .$$Now we have the right to reararray the expression to gain it into a type usable by the Lambert W attributes by separating with by $exp(2 lnx)$ and multiplying by $2$$$Rightarrowhead 2 = -2 ln x e^-2 ln x$$$$ Rightarrow W(2) = -2 ln x$$$$Rightarrowhead x = e^-W(2)/2$$


Thanks for contributing an answer to carolannpeacock.comematics Stack Exchange!

Please be certain to answer the question. Provide details and also share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based upon opinion; ago them up with references or individual experience.

Use carolannpeacock.comJax to format equations. carolannpeacock.comJax recommendation.

See more: Georgia On My Mind Chords Willie Nelson At Countrytabs, Georgia On My Mind Chords

To learn even more, watch our tips on composing great answers.

Article Your Answer Discard

By clicking “Blog post Your Answer”, you agree to our terms of business, privacy plan and cookie plan

Not the answer you're looking for? Browse various other questions tagged lambert-w or ask your own question.


site architecture / logo © 2021 Stack Exadjust Inc; user contributions licensed under cc by-sa. rev2021.9.15.40218

carolannpeacock.comematics Stack Exadjust works best through JavaScript allowed

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your gadget and discshed information in accordance through our Cookie Policy.