1. [latex]log_{frac{1}{2}} 64 = log_{frac{1}{2}} 2^6 = log_{frac{1}{2}} (frac{1}{2})^{-6} = - 6 cdot log_{frac{1}{2}} frac{1}{2} = - 6 cdot 1 = - 6[/latex] [latex]log_2 8sqrt{2} = log_2 2^3 cdot 2^{frac{1}{2}} = log_2 2^{3+frac{1}{2}} = log_2 2^{3frac{1}{2}} = 3frac{1}{2} cdot log_2 2 =\\ =3frac{1}{2} cdot 1= 3frac{1}{2}[/latex] [latex]log_{frac{1}{5}} 1= 0[/latex] [latex]log10=1[/latex] [latex]log_5 25sqrt[3]{5} = log_5 5^2 cdot 5^{frac{1}{3}}=log_5 5^{2+frac{1}{3}}=log_5 5^{2frac{1}{3}}=2frac{1}{3} cdot log_5 5 = \\ =2frac{1}{3} cdot 1 = 2frac{1}{3}[/latex] 2. [latex]log_3 x=-1 \ x = 3^{-1} \ x = frac{1}{3}[/latex] [latex]log_x 25=2 \ x^2 = 25 \ x^2 = 5^2 \ x = 5[/latex] [latex]log_x 64=-3 \ x^{-3} = 64 \ x^{-3} = 4^3 \ x^{-3} = (frac{1}{4})^{-3} \ x = frac{1}{4}[/latex] [latex]log_5 x=3 \ x = 5^3 \ x = 125[/latex] [latex]log_x 3=frac{1}{2} \ x^{frac{1}{2}} = 3 \ x^{frac{1}{2}} = 9^{frac{1}{2}} \ x = 9[/latex]
