[latex]a) log_{frac{1}{2}} 16sqrt[3]{2}=-4frac{1}{3}\ (frac{1}{2})^x=16sqrt[3]{2}\ (2^{-1})^{x}=2^4*2^{frac{1}{3}}\ 2^{-x}=2^{4+frac{1}{3}}\ -x=4frac{1}{3} |*(-1)\ oxed{x=-4frac{1}{3}}[/latex] [latex]b) log_{frac{1}{5}} 5sqrt{5}=-1frac{1}{2}\ (frac{1}{5})^x=5sqrt{5}\ (5^{-1})^x=5^1*5^{frac{1}{2}}\ 5^{-x}=5^{1+frac{1}{2}}\ -x=1frac{1}{2} |*(-1)\ oxed{x=-1frac{1}{2}}[/latex] [latex]c) log_5 125sqrt{5}=3frac{1}{2}\ 5^x=125sqrt{5}\ 5^x=5^3*5^{frac{1}{2}}\ 5^x=5^{3+frac{1}{2}}\ oxed{x=3frac{1}{2}}[/latex] [latex]d) log_{frac{1}{6}} 36sqrt[4]{6}=-2frac{1}{4}\ (frac{1}{6})^x=36sqrt[4]{6}\ (6^{-1})^x=6^2*6^{frac{1}{4}}\ 6^{-x}=6^{2+frac{1}{4}}\ -x=2frac{1}{4} |*(-1)\ oxed{x=-2frac{1}{4}}[/latex] [latex]e) log_{3sqrt{3}} 81sqrt[3]{3}=2frac{8}{9}\ \ (3sqrt{3})^x=81sqrt[3]{3}\ (3^1*3^{frac{1}{2}})^x=3^4*3^{frac{1}{3}}\ (3^{1+frac{1}{2}}})^x=3^{4+frac{1}{3}}\ 3^{1frac{1}{2}x}=3^{4frac{1}{3}}\ 1frac{1}{2}x=4frac{1}{3}\ frac{3}{2}x=frac{13}{3} |*frac{2}{3}\ oxed{x=frac{26}{9}=2frac{8}{9}}[/latex]
