From Förberedande kurs i matematik 1

The logarithm \displaystyle \text{lg }0.\text{1 } is defined as that number which should stand in the (??) grey box in order that the equality

\displaystyle 10^{??}=0.1

should hold. In this case, we see that

\displaystyle 10^{-1}=0.1

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