[tex]log_{\sqrt5}5\sqrt[3]{5}=\frac{log_55\sqrt[3]{5}}{log_5\sqrt5}=\frac{log_5(5\cdot5^{\frac{1}{3}})}{log_55^{\frac{1}{2}}}=\frac{log_55^{1\frac{1}{3}}}{\frac{1}{2}log_55}=\frac{4}{3}\cdot2=\frac{8}{3}=2\frac{2}{3}[/tex]
[tex]\\\\\log_{\frac{1}{5}}25\sqrt5=\frac{log_525\sqrt5}{log_5\frac{1}{5}}=\frac{log_5(5^2\cdot5^{\frac{1}{2}})}{log_55^{-1}}=\frac{log_55^{2\frac{1}{2}}}{-log_5}=\frac{2\frac{1}{2}log_55}{-1}=-2\frac{1}{2}[/tex]
[tex]log_{2\sqrt2}4\sqrt8=\frac{log_24\sqrt8}{log_22\sqrt2}=\frac{log_2(2^2\cdot8^{\frac{1}{2}})}{log_2(2\cdot2^{\frac{1}{2}})}=\frac{log_2(2^2\cdot(2^3)^{\frac{1}{2}})}{log_22^{1\frac{1}{2}}}=\frac{log_2(2^2\cdot2^{\frac{3}{2}})}{1\frac{1}{2}log_22}=\\\\\frac{log_22^{\frac{7}{2}}}{1\frac{1}{2}}=\frac{\frac{7}{2}log_22}{\frac{3}{2}}=\frac{7}{2}\cdot\frac{2}{3}=\frac{7}{3}=2\frac{1}{3}[/tex]
