[tex]a)\\\log_{\frac15}{\sqrt[11]5}=\log_{\frac15}\big5^\frac1{11} = \log_{\frac15}\big(\frac15\big)^{-\frac1{11}}=-\frac1{11}\log_\frac15\frac15=-\frac1{11} \\\\ b) \\ \log_{81}243=\log_{3^4}3^5= \log_{3^4}\left[\big(3^4\big)^\frac14\right]^5 =\log_{3^4} \big(3^4\big)^\frac54\ =\frac54\log_{3^4} 3^4=\frac54\\\\c)\\\sqrt[3]{16}^{\,\log_427}=\left(\big4^\frac43\right)^{\log_427} = \big4^{\frac43\log_427} = \big4^{\log_427^\frac43}=2\big7^\frac43=\big(3^3\big)^\frac43=3^4=81[/tex]
[tex]d)\\\log_a32=-\frac52\quad\iff\quad \big a^{-\frac52}=32\\\\\big a^{-\frac52}=32\qquad\big/^{-\frac25}\\\\a=3\big2^{-\frac25}=\left(2^5\right)^{-\frac25}=\big2^{-2}=\big(\frac12\big)^2=\frac14[/tex]
