From Förberedande kurs i matematik 1

The logarithm \displaystyle \mathop{\text{lg}} 0\textrm{.}1 is defined as that number which should stand in the coloured box in order that the equality

\displaystyle 10^{\bbox[#FFEEAA;,1.5pt]{\phantom{\scriptstyle ??}}} = 0\textrm{.}1

should hold. In this case, we see that

\displaystyle 10^{-1} = 0\textrm{.}1

and therefore \displaystyle \mathop{\text{lg}} 0\textrm{.}1 = -1\,.