Answer
$\dfrac{2}{3}$
Work Step by Step
Using $\log_bx^m=m\log_bx$ or the Power Rule of Logarithms, the given expression, $
\log_{1000}100
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_{1000}1000^{2/3}
\\\\=
\dfrac{2}{3}\log_{1000}1000
.\end{array}
Since $\log_bb=1,$ the expression, $
\log_{1000}1000
,$ simplifies to $1$.
\begin{array}{l}\require{cancel}
\dfrac{2}{3}(1)
\\\\=
\dfrac{2}{3}
.\end{array}