I'm going through some problems in Theorems, Corollaries, Lemmas, and Methods of Proof, and I'm stuck at a certain problem that seemed very interesting until I couldn't solve it for the life of me.
Let $\circ$ be defined on $\mathbb R^+$ as $a\circ b = a^b b^a$. One can show that $\circ$ is Abelian, that $\mathbb R^+$ is closed under $\circ$, and that there exists identity element $e=1$ such that $a\circ e = e \circ a = a$.
The next part of the problem asks to find $2^{-1}$, which means solving for $x$ where $2^x x^2 = 1$. Because $\circ$ is Abelian, there's no other way to arrange this expression. I tried taking logarithm of both sides to get
$\begin{align} \log_2(2^x x^2) &= \log_2(1) \\ \log_2(2^x) + \log_2(x^2) &= 0 \\ x + 2\log_2(x) &= 0 \\ x &= -2\log_2(x) \\ 1 &= -2 \frac{\log_2(x)}{x} \\ -\frac{1}{2} &= \log_2(x^{\frac{1}{x}}) \\ 2^{-\frac{1}{2}} &= x^{\frac{1}{x}} \end{align}$
This doesn't really get me anywhere. I tried other manipulations, but all expressions I got were quite ugly. WolframAlpha doesn't return any meaningful results for positive reals. I keep coming back to this problem, but it looks like I won't be able to solve it. Perhaps I'm missing some algebra I have to use to get the result.