Domain Problem: $\sqrt{ \log_{\frac{1}{2}} x}$

Question

How to find the domain of the function $\sqrt{ \log_{\frac{1}{2}} x}$ ?

Answer 1

Assuming the result is real, we must have $\log _{\frac{1}{2}}x\geq 0$. Since $\log _{\frac{1}{2}}x\geq 0\Leftrightarrow 0<x\leq 1$, the domain is $% 0<x\leq 1$, with $x\in \mathbb{R}$.

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Added 2: $\log _{\frac{1}{2}}x=\frac{\log x}{\log \frac{1}{2}}=\frac{\log x}{\log 1-\log 2}=\frac{\log x}{0-\log 2}=-\frac{\log x}{\log 2}$

$\log _{\frac{1}{2}}x\geq 0\Leftrightarrow-\frac{\log x}{\log 2}\geq 0\Leftrightarrow\log x\le 0\Leftrightarrow 0<x\leq 1.$

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Added: plot of $\log_{\frac{1}{2}}x$ (green) and $\sqrt{\log_{\frac{1}{2}}x}$ (blue).