<s>[INST] How can I find the domain of this sqrt log function? 
Can someone help me to find the domain of this function step by steps,
I know how to find the domain for a log by making it >0 but I am unsure about the process doing it with 2 different logarithm and a square root.
$$f(x)=\sqrt{\log_{0.5}(5-x)+\log_2(2x-4)-1}$$

 [/INST] Let's first define both the $\log$s, input of $\log$ should be a positive quantity.
$5 - x > 0 \implies x \in (-\infty, 5)$

$2x - 4 > 0 \implies x \in (2, \infty)$

Now we know that $\log_{a}{b} = -\log_{\frac{1}{a}}{b} = \log_{\frac{1}{a}}{\frac{1}{b}}$
$\implies\log_{0.5}(5-x) = \log_{2}{\frac{1}{5-x}}$
Now, we know that anything inside square root should be non negative.
$\implies \log_{0.5}(5-x)+\log_2(2x-4)-1 \geq 0$
$\implies \log_{2}{\frac{1}{5-x}+\log_2(2x-4)-1} \geq 0$
$\implies \log_2{\frac{2x-4}{5-x}} \geq 1$
$\implies \frac{2x-4}{5-x} \geq 2$
$\implies \frac{2x-4}{5-x} - 2 \geq 0$
$\implies \frac{4x-14}{5-x} \geq 0$
$\implies \frac{4x-14}{x-5} \leq 0$
$\implies x \in [\frac72, 5)$

Taking intersection of all 3 intervals:
$$\boxed{x \in [\frac72,5)}$$
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