Tính giá trị của các biểu thức sau:

a) ${\log }_8\frac{1}{32}$ ;                                                

b) log₅ 3 . log₃ 5;

c) $2^{\frac{1}{{\log }_{5}2}}$ ;                                           

d) log₂₇ 25 . log₅ 81.

a) ${\log }_8\frac{1}{32}=\frac{{\log }_2\frac{1}{32}}{{\log }_{2}8}=\frac{{\log }_2{2}^{-5}}{{\log }_2{2}^3}=-\frac{5}{3}$ ;

b) ${\log }_{5}3.{\log }_{3}5={\log }_{5}3\frac{1}{{\log }_{5}3}=1$ ;

c) $2^{\frac{1}{{\log }_{5}2}}=2^{lo{g}_{2}5}=5$ ;

d) ${\log }_{27}25.{\log }_{5}81=\frac{{\log }_{3}25}{{\log }_{3}27}\cdot \frac{{\log }_{3}81}{{\log }_{3}5}$

$=\frac{{\log }_3{5}^2}{{\log }_3{3}^3}\cdot \frac{{\log }_3{3}^4}{{\log }_{3}5}=\frac{2{\log }_{3}5}{3}\cdot \frac{4}{{\log }_{3}5}$$=\frac{8}{3}$