[latex]log_{sqrt{5}}5=log_{sqrt{5}}(sqrt{5})^2=2log_{sqrt{5}}sqrt{5}=2cdot 1=2\ log_{sqrt{3}}27=log_{sqrt{3}}(sqrt{3})^6=6log_{sqrt{3}}sqrt{3}=6cdot 1=6\ log_{sqrt{2}}4sqrt{2}=log_{sqrt{2}}4+log_{sqrt{2}}sqrt{2}=log_{sqrt{2}}(sqrt{2})^4+1=4+1=5\ log_{ sqrt[3]{6} }6=log_{ sqrt[3]{6} }(sqrt[3]{6} )^3=3log_{ sqrt[3]{6} }(sqrt[3]{6} )=3cdot 1=3[/latex] [latex] log_{ sqrt[3]{10} }sqrt{10}=log_{ sqrt[3]{10} }10^frac{1}{2}=dfrac{1}{2}log_{ sqrt[3]{10} }10=dfrac{1}{2}log_{ sqrt[3]{10}}( sqrt[3]{10} )^3=\ =dfrac{1}{2}cdot 3cdot 1=dfrac{3}{2}\ log_{sqrt{7}} sqrt[3]{49} =log_{sqrt{7}} (49)^{frac{1}{3}}=dfrac{1}{3}log_{sqrt{7}}49=dfrac{1}{3}log_{sqrt{7}}7^2=\ =dfrac{2}{3}log_{sqrt{7}}7=dfrac{2}{3}log_{sqrt{7}}(sqrt{7})^2=dfrac{4}{3}log_{sqrt{7}}sqrt{7}=dfrac{4}{3}cdot 1=dfrac{4}{3}[/latex]
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