Aus Online Mathematik Brückenkurs 1

First, we rewrite the number 0.125 as a fraction which we also simplify

\displaystyle 0\textrm{.}125 = \frac{125}{1000} = \frac{5\cdot 25}{10^3} = \frac{5\cdot 5\cdot 5}{(2\cdot 5)^3} = \frac{1}{2^3} = 2^{-3}\,\textrm{.}

Because 0.125 was expressed as a power of 2, the logarithm can be calculated in full

\displaystyle \log_2 0\textrm{.}125 = \log_2 2^{-3} = (-3)\cdot\log_2 2 = (-3)\cdot 1 = -3\,\textrm{.}