a)
log_{\sqrt5}5\sqrt[3]{5}=log_{\sqrt5}5*5^{\frac{1}{3}}=log_{\sqrt5}5^{\frac{15+1}{3}}=log_{\sqrt5}5^{\frac{16}{3}}=log_{\sqrt5}(5^{\frac{1}{2}})^{\frac{32}{3}}=log_{\sqrt5}(\sqrt5)^{\frac{32}{3}}=\frac{32}{3}=10\frac{2}{3}
b)
log_{\sqrt[3]{3}}27=log_{\sqrt[3]{3}}3^3=log_{\sqrt[3]{3}}(3^{\frac{1}{3}})^9=log_{\sqrt[3]{3}}(\sqrt[3]{3})^9=9
c)
log_28\sqrt2=log_22^3*2^{\frac{1}{2}}=2^{\frac{6+1}{2}}=2^{\frac{7}{2}}=\frac{7}{2}=3,5
d)
log_{\frac{1}{3}}81\sqrt3=log_{\frac{1}{3}}(3^4)*3^{\frac{1}{2}}=log_{\frac{1}{3}}3^{\frac{8+1}{2}}=log_{\frac{1}{3}}3^{\frac{9}{2}}=log_{\frac{1}{3}}(\frac{1}{3})^{-\frac{9}{2}}=-\frac{9}{2}=-4,5