[tex]\displaystyle log_{27}25 \cdot log_49 \cdot log_58 \\ \\ log_{27}25= \frac{log_425}{log_427} = \frac{log_45^2}{log_43^3} = \frac{2log_45}{3log_43} \\ \\ log_58= \frac{log_48}{log_45} = \frac{log_44^ \frac{3}{2} }{log_45} = \frac{ \frac{3}{2}log_44 }{log_45} = \frac{ \frac{3}{2} }{log_45} = \frac{3}{2} :log_45= \frac{3}{2} \cdot \frac{1}{log_45} = \frac{3}{2log_45} [/tex]
[tex]\displaystyle log_{27}25 \cdot log_49 \cdot log_58= \frac{2log_45}{3log_43} \cdot log_49 \cdot \frac{3}{2log_45} = \\ \\ = \frac{2 \cdot3log_45 \cdot log_49}{2 \cdot 3log_43 \cdot log_45} = \frac{log_49}{log_43}= \frac{log_43^2}{log_43} = \frac{2log_43}{log_43} =\boxed{2}[/tex]
