# Does $^{\frac12}x=e^{W(\ln x)}$, or not? [duplicate]

1. If we restrict ourselves to positive real values, $^{\frac12}x=e^{W(\ln x)}$ is the only consistent definition for $^{\frac12}x$; and
2. This definition is sufficient to define tetration to any real height, if we further say that $\displaystyle^tx = {\lim_{a \to t}}\; ^ax$. So, what gives? I don't especially think I'm smarter than professional mathematicians, so if there's an inconsistency I'd like to be able to recognize it. Is there?
• Could you expand on how you would define ${}^tx$ using only integer values and the fact that $^{\frac12}x=e^{W(\ln x)}$? – Omnomnomnom Aug 10 '13 at 18:43
• so $e^{W(W(\ln(x)))}=e^x$ ? Try W|A wolframalpha.com/input/?i=e^[W[W[ln[x]]]] – Gottfried Helms Aug 20 '13 at 20:27
• How are we supposed to know this given we can't really evaluate $^{1/2}x$ well? – Simply Beautiful Art Dec 31 '15 at 17:03