How do you solve x^x = c for x if c is a constant?

Answer 1

There is no general solution to this problem in terms of elementary functions.

The solution to this equation can, however, be expressed in terms of the Lambert W function (see source) as:

x = (ln(c))/W(ln(c))

where W(z) is the Lambert W function, which is defined as the inverse function of f(W) = W*exp(W).

Answer 2

first you convert to exponential form.
i.e. e^(xlnx) = c
xlnx = lnc

This equation can't be solved exactly (it is a transcendental equation), but you can solve it numerically. Given c you can perform the appropriate expansion of lnx as either a taylor or a harmonic series and then solve it as a polynomial.
Ultimately it's a question for a computer. Unless c is a value that makes it obvious what the answer is (like 0, 1, 4, 27 etc), you won't be able to solve it exactly.

Answer 3

You can't; you can only solve it using approximation methods.

Just like you can't solve x - sinx = 5, or lnx + x = 33, you can't solve x^x = c.

x^x = c
xlnx = lnc

and then from here we can get nowhere.

Answer 4

i would this the best way would be guess and check.. it would be pretty easy to make an initial guess and then work off of that, raising or lowering it as needed