# Hey, how do I solve this? sqrt(x^(logsqrt(x)))=10

Mar 26, 2017

$x = {e}^{\pm \sqrt{4 \log \left(10\right)}}$

#### Explanation:

$\sqrt{{x}^{\log \left(\sqrt{x}\right)}} = 10$ squaring

${x}^{\log \left(\sqrt{x}\right)} = {10}^{2}$

applying $\log$ to both sides

$\log \left(\sqrt{x}\right) \log x = 2 \log 10$ or

$\frac{1}{2} {\left(\log x\right)}^{2} = 2 \log 10$

then

$\log x = \pm \sqrt{4 \log \left(10\right)}$

and finally

$x = {e}^{\pm \sqrt{4 \log \left(10\right)}}$

NOTE:

Adopting $\log \left(x\right) \equiv {\log}_{10} x$ the result will be

$x = {10}^{\pm \sqrt{4 {\log}_{10} 10}} = {10}^{\pm 2} = \left\{\begin{matrix}0.01 \\ 100\end{matrix}\right.$