Ilya Zakharevich on Tue, 21 Nov 2023 04:50:50 +0100

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  gp > ?lambertw
  lambertw(y,{branch=0}): solution of the implicit equation
  x*exp(x)=y. In the p-adic case, gives a solution of x*exp(x)=y if x
  has positive valuation, of x+log(x)=log(y) otherwise.

First, I would rewrite it as log(x)+x = log(y) to make the match more
clear.  Second, I suspect that what is meant is

   y has positive valuation

(not “x”).

Hope this helps,