From Förberedande kurs i matematik 1

The argument in the logarithm can be rewritten as \displaystyle \frac{1}{10^{2}} = 10^{-2} and then the log law \displaystyle \lg a^b = b\lg a gives the rest

\displaystyle \lg \frac{1}{10^2} = \lg 10^{-2} = (-2)\cdot \lg 10 = (-2)\cdot 1 = -2\,\textrm{.}