[tex]log_{\frac{1}{5}}125 = log_{\frac{1}{5}}5^3 = log_{\frac{1}{5}}((\frac{1}{5})^{-3}) = -3\\log_{2}4\sqrt{2} = log_{2}(2^2 * 2^\frac{1}{2}) = log_{2}2^{2\frac{1}{2}} = 2\frac{1}{2}\\log_{\sqrt{5}}(log_{3}243) = log_{\sqrt{5}}(log_{3}3^5) = log_{\sqrt{5}}(5) = log_{\sqrt{5}}(\sqrt{5}^2) = 2\\log_{5}25 + log_{4}\frac{1}{8} = log_{5}5^2 + log_{4}\frac{1}{4^\frac{3}{2}} = 2 + log_{4}4^{-\frac{3}{2}}} = 2 - \frac{3}{2} = \frac{1}{2}\\[/tex]
[tex]\frac{log4 - log\frac{2}{5} }{2*log_{6}2+log_{6}9} = \frac{log(4 * \frac{5}{2} )}{log_{6}(4*9)} = \frac{log10}{log_{6}36} = \frac{1}{2}[/tex]
