[tex]log_3 \ 3 = 1 \\\\ log_3 \ \frac{1}{9} =log_3 \ 9^{-1} =log_3 \ (3^2)^{-1} =log_3 \ 3^{-2} = -2 \cdot log_3 \ 3 = -2 \cdot 1=-2 \\\\ log_3 \ 1 = 0 \\\\ log_3 \ \sqrt{3} =log_3 \ 3^{\frac{1}{2}}=\frac{1}{2} \cdot log_3 \ 3 = \frac{1}{2} \cdot1=\frac{1}{2} \\\\ log_2 \ 8 =log_2 \ 2^3 = 3 \cdot log_2 \ 2 = 3 \cdot 1 =3 \\\\ log_{\sqrt{2}} \ 8 =log_{\sqrt{2}} \ 2^3 =3 \cdot log_{\sqrt{2}} \ 2= 3 \cdot log_{\sqrt{2}} \ (\sqrt{2})^2 = 3 \cdot 2 \cdot log_{\sqrt{2}} \ \sqrt{2} = 6 \cdot 1 = 6[/tex]
[tex]log_{\frac{1}{2}} \ 8 = log_{\frac{1}{2}} \ 2^3=3 \cdot log_{\frac{1}{2}} \ 2=3 \cdot log_{\frac{1}{2}} \ (\frac{1}{2})^{-1} = 3 \cdot (-1) \cdot log_{\frac{1}{2}} \ \frac{1}{2} = \\ =-3 \cdot 1 = -3 \\\\ log_4 \ 8 = log_4 \ (4 \cdot 2) = log_4 \ 4 + log_4 \ 2 = 1 + log_4 \ \sqrt{4} = 1 + log_4 \ 4^{\frac{1}{2}}= \\ = 1 + \frac{1}{2} \cdot log_4 \ 4 = 1 + \frac{1}{2} \cdot 1 = 1+\frac{1}{2} = 1\frac{1}{2}[/tex]
