I was thinking of $\frac{4}{3}$ and found that $(\frac{4}{3})^4$ roughly equals $3$ (very roughly), and I though I would try to find the pairs of numbers where the equality suffices. So given the equation
$(\frac{a}{b})^a = b$,
how does one rearranges the sides so that the Lambert $W$ function becomes somehow applyable? Or am I getting it completely wrong and it's not the $W$ function but something completely different that is needed in cases like this?
Let $a=be^x$
$b=(\frac{a}{b})^a$
$\ln(b)=a\ln(\frac{a}{b})=a\ln(e^x)=ax=be^xx$
$xe^x=\frac{\ln(b)}{b}$
$x=W(\frac{\ln(b)}{b})$
$a=be^{W(\frac{\ln(b)}{b})}$