# Solution(s) to 'power equations'

I'm not sure a 'power equation' is the right name for the equation I'd like to know more about (and, specifically, about its solutions), but I don't know the 'proper' way to name it.

The simplest 'power polynomial' is : $P(x) = x^x$. The simplest 'power equation is: (1) $x^x = c$ for some $c \in \mathbb{R}$. What are the exact solutions to (1) of x (in $\mathbb{C}$) with regards to c, apart from the 'obvious' solutions (x=0, c=0), (x=1, c=1) and (x=2, c=4) and all the other solutions of the form $x^x = n^n$ for $n \in \mathbb{N}$ ?

We could extend the power polynomial: $P_{2}(x) = x^{{ax}^{bx}} + x^{cx}$. What are the exact solutions of $P_{2}(x) = d$ for $a,b,c,d \in \mathbb{R}, x\in \mathbb{C}$ ? Or perhaps I should ask what the exact form of the solutions is.

We could generalize the power polynomial even further to $P_{3}(x)$ and $P_{n}(x)$, but I don't know how to write down the latter, general polynomial.

Max

(P.S. I References are always welcome. II If you think this question belongs to MO, please tell me).

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Lambert's W en.wikipedia.org/wiki/Lambert%27s_W_function is a start... but beyond there I'm stumped... – Tom Boardman Aug 10 '10 at 13:23
@ Tom and MRA: thanks, this a good start indeed. – Max Muller Aug 10 '10 at 13:26
There is no reason to expect that such equations have solutions in terms of familiar functions. – Qiaochu Yuan Aug 10 '10 at 16:45
There is also no reason to expect that anyone has ever studied this problem. These kinds of equations don't, to my knowledge, appear naturally in any problem. – Qiaochu Yuan Aug 10 '10 at 17:08
x=0, c=0 isn't an "obvious solution." The expression $0^0$ is indeterminate. If anything, you should use $0^0=1$ here since $\lim_{x\rightarrow 0} x^x = 1$. – Corey Jun 27 '11 at 2:32

In regards to the first solution problem, $x^x=c$, you can isolate the $x$ using the Lambert W function as follows:$$x^x=c$$$$x=c^{1/x}$$$$1=\frac1xc^{\frac1x}=\frac1xe^{\ln(c)\frac1x}$$$$\ln(c)=\ln(c)\frac1xe^{\ln(c)\frac1x}$$$$W(\ln(c))=\ln(c)\frac1x$$$$x=\frac{\ln(c)}{W(\ln(c)}=e^{W(\ln(c))}$$

For higher power towers, we require special conditions or situations for a solution to exist in a sort of closed form using the Lambert W function.

Which means that if we can't isolate $x$ in a simple higher power tower like $$x^{x^x}=c$$Then we probably can't do it for your situation.

I will also note that a solution to $$x^{x^{x^{\cdots x}}}=c$$$$x=\sqrt[c]{c}$$

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These types of object are called super(hyper) polynomials which are polynomials of power towers or tetrations. You may want to look at http://en.wikipedia.org/wiki/Tetration for more information and references. These are fairly complicated structures and an object of current research.

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Maybe you should read about the Lambert W-function which gives the solutions to expressions like $z=x^x$. However, I am not sure what to do in case of such a "power tower" like $P_2$.

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