${\log }_{a}b=c\text{, gdy }{a}^c=b$

$a>0,a\ne 1,b>0,c\in \mathbb{R}$

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a)

${\log }_{2}1=0\text{, bo }{2}^0=1$   

${\log }_{2}4=2\text{, bo }{2}^2=4$  

${\log }_2\sqrt{2}=\frac{1}{2}\text{, bo }{2}^{\frac{1}{2}}=\sqrt{2}$

${\log }_{2}0,125={\log }_2\frac{1}{8}=-3\text{, bo }{2}^{-3}=\frac{1}{8}=0,125$

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b)

${\log }_{\frac{1}{2}}2=-1\text{, bo }{\left(\frac{1}{2}\right)}^{-1}=2$

${\log }_{\frac{1}{2}}4=-2\text{, bo }{\left(\frac{1}{2}\right)}^{-2}=2^2=4$   

${\log }_{\frac{1}{2}}\sqrt{2}=-\frac{1}{2}\text{, bo }{\left(\frac{1}{2}\right)}^{-\frac{1}{2}}=2^{\frac{1}{2}}=\sqrt{2}$

${\log }_{\frac{1}{2}}16=-4\text{, bo }{\left(\frac{1}{2}\right)}^{-4}=2^4=16$   

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c)

$\log 100=2\text{, bo }{10}^2=100$

$\log \sqrt{10}=\frac{1}{2}\text{ bo }{10}^{\frac{1}{2}}=\sqrt{10}$

$\log 0,01=-2\text{, bo }{10}^{-2}={0,1}^2=0,01$

$\log \sqrt[3]{10}=\frac{1}{3}\text{, bo }{10}^{\frac{1}{3}}=\sqrt[3]{10}$

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d)

${\log }_{125}\frac{1}{5}=-\frac{1}{3}\text{ bo }{125}^{-\frac{1}{3}}={\sqrt[3]{125}}^{-1}=5^{-1}=\frac{1}{5}$

${\log }_{\frac{1}{5}}125=-3\text{, bo }{\left(\frac{1}{5}\right)}^{-3}=5^3=125$

${\log }_{\frac{1}{\sqrt{5}}}\sqrt[3]{5}=-\frac{2}{3}\text{, bo }{\left(\frac{1}{\sqrt{5}}\right)}^{-\frac{2}{3}}={\sqrt{5}}^{\frac{2}{3}}=5^{\frac{1}{3}}=\sqrt[3]{5}$

${\log }_{5}0,04={\log }_5\frac{1}{25}=-2\text{, bo }{5}^{-2}=\frac{1}{25}=0,04$