$\text{a)}{\log }_{3}x=-1$  

$x=3^{-1}=\frac{1}{3}$     

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$\text{b)}{\log }_{5}x=3$  

$x=5^3=125$

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$\text{c)}{\log }_{\frac{1}{2}}x=-2$  

$x={\left(\frac{1}{2}\right)}^{-2}=2^2=4$  

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$\text{d)}{\log }_{\frac{1}{3}}x=-\frac{1}{2}$  

$x={\left(\frac{1}{3}\right)}^{-\frac{1}{2}}=3^{\frac{1}{2}}=\sqrt{3}$  

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$\text{e)}{\log }_{2}x=-\frac{2}{3}$  

$x=2^{-\frac{2}{3}}={\left(\frac{1}{2}\right)}^{\frac{2}{3}}=\sqrt[3]{{\left(\frac{1}{2}\right)}^2}=\sqrt[3]{\frac{1}{4}}=\frac{1}{\sqrt[3]{4}}$  

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$\text{f)}{\log }_{4}x=0$  

$x=4^0=1$  

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$\text{g)}{\log }_{2}x=10$  

$x=2^{10}=1024$  

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$\text{h)}{\log }_{2\sqrt{2}}x=-3$  

$x={\left(2\sqrt{2}\right)}^{-3}={\left(2^{\frac{3}{2}}\right)}^{-3}=2^{-\frac{9}{2}}=2^{\frac{1}{2}-5}=\frac{2^{\frac{1}{2}}}{2^5}=\frac{\sqrt{2}}{32}$