1. Oblicz: a) [latex]log_{ { sqrt{5} } }5 sqrt{5}[/latex] = b) [latex]log x_{ sqrt[3]{3} } 27[/latex] = c) [latex]log_{2}8 sqrt{2} [/latex] = d) [latex]log_{ frac{1}{3} }81sqrt{3} [/latex] =

a) [latex]log_{ { sqrt{5} } }5 sqrt{5}=log_{ {5^{ frac{1}{2} }} } 5 cdot 5^{ frac{1}{2} }=2log_55^{ frac{3}{2}}=2 cdot frac{3}{2}log_55=3[/latex] b) [latex]log_{ sqrt[3]{3} } 27 =log_{3^{ frac{1}{3} }} 3^3=3log_{3} 3^3=3 cdot 3log_{3} 3=9[/latex] c) [latex]log_{2}8 sqrt{2} =log_{2}(2^3 cdot 2^{ frac{1}{2} })=log_{2} 2^{ frac{7}{2}}=frac{7}{2}log_22=frac{7}{2}=3,5[/latex] d) [latex]log_{ frac{1}{3} }81sqrt{3}=log_{ frac{1}{3} }(3^4 cdot 3^{ frac{1}{2} })=log_{ frac{1}{3} }3^{ frac{9}{2} }=log_{ frac{1}{3} }left( frac{1}{3} ight) ^{ -frac{9}{2} }= -frac{9}{2}log_{ frac{1}{3} }frac{1}{3}=-frac{9}{2}=-4,5[/latex]