exact solution to $x+x^x=3$

Question

I'm trying to find the exact solution to the equation $x+x^x=3$. I know the answer is approximately 1.4, but what is its formal definition? I understand it may not be from the result of a real function, since the exact solution to its brother, $x^x=3$, requires use of the Lambert W function for its formal definition ($x=e^{W(\ln 3)} $), which cannot be expressed in elementary terms. I have tried looking this equation up on Wolfram alpha but it can only approximate $x$. What would be the exact definition of this variable?

Answer 1

Irrational numbers are usually defined as solutions of equations, for example $x^2=2$, as limits, say $\sum_{k=0}^\infty(-1)^k\frac{4}{2k+1}$ or $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{1/n}$, or by introducing new notation, say $\sqrt{2},\pi$ or $e$. You stated the definition as a solution of an equation. Since introducing new notation is not helpful in this case, I only propose the following two alternatives. Let $x^*\in(0,\infty)$ be the solution of $x+x^x=3$.

1. Limit: Consider the sequence given by $a_0=1$ and $a_{k+1}=\frac{3}{1+a_k^{a_k-1}}$ for $k\ge 0$. The sequence $(a_k)_k$ converges to the solution of the equation, i.e. $x^*=\lim_{k\rightarrow\infty}a_k$.
2. Using the inverse function: Let $f:[1,\infty)\rightarrow(0,\infty)$, $x\mapsto x+x^x$, and notice that $f$ is strictly increasing since $f'(x)=1+(\ln(x)+1)x^x>2$, so the solution of the equation is $x^*=f^{-1}(3)$.

These would be three common ways (equation, limit, function value) to formally define the number $x^*$. I hope this helps!