Lambert- W -Function calculation? I have an equation of the form: n log n=x Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for it's evaluation. Any ideas as to how I can evaluate?

Davirnoilc

Davirnoilc

Answered question

2022-11-15

Lambert- W -Function calculation?
I have an equation of the form:
n log n = x
Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for it's evaluation.
Any ideas as to how I can evaluate?

Answer & Explanation

ontzeidena8a

ontzeidena8a

Beginner2022-11-16Added 17 answers

Although an old post, I'm surprised nobody mentioned this:
n log n = ( e log n ) log n = x
Now it should be obvious on how to proceed.
W ( x ) = log n
e W ( x ) = n
And because W ( x ) e W ( x ) = x, e W ( x ) = x W ( x ) so,
x W ( x ) = n
Hanna Webster

Hanna Webster

Beginner2022-11-17Added 3 answers

Although this has already been answered, I think a step by step explanation is more informative than a general one.
The following is my attempt:
n ln ( n ) = x ln n n = x n n = e x n = e x / n n × x n = x n e x / n x = x n e x / n W ( x ) = W ( x n e x / n ) W ( x ) = x n n = x W ( x ) n = x x / e W ( x ) (since  W ( x ) e W ( x ) = x ) n = e W ( x )

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