Solve in R

**dhiab** · August 17th 2009, 08:04 PM

Solve in R : $\left( {x^x } \right)^2 = 8$

**pickslides** · August 17th 2009, 08:48 PM

Interesting question, I said $x^x = \sqrt{8}$ then using trial and error I found $x \approx 1.788454$

**Random Variable** · August 17th 2009, 08:59 PM

$x^{x}= \sqrt{8} = 2^{3/2}$

$e^{\ln x \ e^{\ln x}} = 2^{3/2}$

$\ln (e^{\ln x \ e^{\ln x}}) = \ln (2^{3/2})$

$\ln x e^{\ln x} =\frac{3}{2} \ln 2$

$\ln x = \frac{\frac{3}{2}\ln 2}{e^{\ln x}} = W\Big(\frac{3}{2} \ln 2\Big)$

where W is the Lambert W function

so $x = e^{W\Big(\frac{3}{2} \ln 2\Big)} \approx 1.788$

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