Answer
false
Work Step by Step
Using the properties of logarithms, the left-hand expression, $
\dfrac{\log_{10}10}{\log_{10}100}
$, is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{\log_{10}10}{\log_{10}10^2}
\\\\&=
\dfrac{\log_{10}10}{2\log_{10}10}
&(\text{use }\log_b x^y=y\log_b x)
\\\\&=
\dfrac{\cancel{\log_{10}10}}{2\cancel{\log_{10}10}}
\\\\&=
\dfrac{1}{2}
.\end{align*}
Thus, the left-hand expression is NOT equivalent to the given right-hand expression, $\dfrac{1}{10}$. Hence, the given statement is false.