From Förberedande kurs i matematik 1

We know that \displaystyle 10^{\lg x}=x, so therefore we rewrite the exponent as \displaystyle -\lg 0.1=\left( -1 \right)\centerdot \lg 0.1=\lg 0.1^{-1} by using the log law \displaystyle b\lg a=\lg a^{b}. This gives

\displaystyle 10^{-\lg 0.1}=10^{\lg 0.1^{-1}}=0.1^{-1}=\frac{1}{0.1}=10