From Förberedande kurs i matematik 1

First, we rewrite the number \displaystyle 0.\text{125 } as a fraction which we also simplify:

\displaystyle 0.\text{125 }=\frac{\text{125 }}{1000}=\frac{5\centerdot 25}{10^{3}}=\frac{5\centerdot 5\centerdot 5}{\left( 2\centerdot 5 \right)^{3}}=\frac{1}{2^{3}}=2^{-3}

Because \displaystyle 0.\text{125 } was expressed as a power of \displaystyle \text{2}, the logarithm can be calculated in full:

\displaystyle \log _{2}0.\text{125 }=\log _{2}2^{-3}=\left( -3 \right)\centerdot \log _{2}2=\left( -3 \right)\centerdot 1=-3