Question

If $\frac{1}{2} \leq \log_{0.1} x \leq 2$ then

Easy

Solution

$\frac{1}{2} \leq \log_{0.1} x \leq 2 \frac{1}{2} \leq \log_{0.1} x \Rightarrow \log_{0.1} {(0.1)}^{1 / 2} \leq \log_{0.1} x \Rightarrow {(0.1)}^{1 / 2} \geq x \Rightarrow x \leq \frac{1}{\sqrt{10}} \log_{0.1} x \leq 2 \Rightarrow \log_{0.1} x \leq \log_{0.1} {(0.1)}^{2} x \geq {(0.1)}^{2} \Rightarrow x \geq \frac{1}{100}, ∴ \frac{1}{100} \leq x \leq \frac{1}{\sqrt{10}} . Hence, x_{\max} = \frac{1}{\sqrt{10}}, x_{\min} = \frac{1}{100} .$

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