Answer
$-\dfrac{1}{2}$
Work Step by Step
Note that $0.1 = \dfrac{1}{10}$.
The given expression is equivalent to:
$=\log_{100}{(\frac{1}{10})}$
Note that $10=\sqrt{100} = 100^{\frac{1}{2}}$.
Thus, the expression above is equivalent to:
$=\log_{100}{\left(\frac{1}{100^{\frac{1}{2}}}\right)}$
Use the rule $\dfrac{1}{a^m} = a^{-m}$ to obtain:
$=\log_{100}{(100^{-\frac{1}{2}})}$
RECALL:
$\log_b{(b^{x})}=x$
Use the rule above with $b=100$ and $x=-\frac{1}{2}$ to obtain:
$=-\dfrac{1}{2}$
