From Förberedande kurs i matematik 1

If we write \displaystyle \text{4} and \displaystyle \text{16} as

\displaystyle \begin{align} & \text{4}=2\centerdot 2=2^{2} \\ & 16=2\centerdot 8=2\centerdot 2\centerdot 4=2\centerdot 2\centerdot 2\centerdot 2=2^{4} \\ \end{align}

we obtain

\displaystyle \begin{align} & \log _{2}4+\log _{2}\frac{1}{16}=\log _{2}2^{2}+\log _{2}\frac{1}{2^{4}} \\ & =\log _{2}2^{2}+\log _{2}2^{-4}=2\centerdot \log _{2}2+\left( -4 \right)\centerdot \log _{2}2 \\ & =2\centerdot 1+\left( -4 \right)\centerdot 1=-2 \\ \end{align}