How is the x^x equation solved?

Question

Mathematics

Dear Math experts, please remind the bydlocoder how equations like
x^x = c are solved, where c is a known constant
. It suddenly became clear that the equation of the species nlogn = 1000introduces individual characters into a stupor.

Answer 1

The equation x*ln(x)=c is easier to solve. You can use Newton's method. And iterations are also possible: x_{n+1}=c/ln(x_n), it should converge very quickly.
UPD. Although no, it will only converge for c>e. Newton's method is still more reliable:
x_{n+1}=(c+x_n)/(ln(x_n)+1)

Answer 2

Newton's method (or tangents) will be categorically faster than half division, despite the fact that the code will be quite a bit more complicated.

Answer 3

Thanks a lot. Terribly good comments. It is a pity that you can not put 2 checkboxes "Solution" - @thousandsofthem's answer seemed to me the simplest (at least understandable to me). I have no doubt that @Mrrl 's answer is also great, but Newton's method, alas, still needs to be raised. Numerical methods are completely forgotten :( why was it not possible to study normally at the institute?