Solving \(\displaystyle{8}{n}^{{2}}={64}{n}{\log{{\left({n}\right)}}}\)

Lucian Ayers

Lucian Ayers

Answered question

2022-04-02

Solving 8n2=64nlog(n)

Answer & Explanation

Janessa Foster

Janessa Foster

Beginner2022-04-03Added 12 answers

Your equation does not have a closed form in terms of standard functions you come across in school, but it does have a solution in terms of the Lambert W-function W(z). The defining equation for W(z) is
z=W(z)eW(z)
We want to rearrange your equation into something similar so we get
n=8logn
8n=logn
en8=n
1=nen8
18=n8en8
Now
Y=XeXX=W(Y)
Thus we get
n8=W(18)
n=8W(18)
To get decimal approximations, you can numerically evaluate W for its different branches and you get the solutions wolfram alpha gave you.

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