Kano Logarithms

$\begin{aligned} \log_\frac 12 16 - \log_9 27 & = \frac {\log 16}{\log \frac 12} - \frac {\log 27}{\log 9} \\ & = \frac {\log 2^4}{-\log 2} - \frac {\log 3^3}{\log 3^2} \\ & = - \frac {4\log 2}{\log 2} - \frac {3\log 3}{2\log 3} \\ & = - 4 - \frac 32 = \boxed {-5.5} \end{aligned}$

$\log_{1/2} 16 - log _9 27$

$= \log_{1/2} {1/2}^{-4} - log_9 9^{1.5}$

$=-4\log_{1/2} {1/2} - 1.5\log_9 9$

$= -4 - 1.5$

$=-5.5$