#### Answer

$-0.6532$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
Use the Laws of Logarithms to find an equivalent expression for the given expression, $
\log_{10} \dfrac{2}{9}
,$ which uses the given logarithmic values, $\log_{10} 2=
0.3010
$ and/or $\log_{10} 3=
0.4771
.$
$\bf{\text{Solution Details:}}$
The given expression can be expressed as
\begin{array}{l}\require{cancel}
\log_{10} \dfrac{2}{3^2}
.\end{array}
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\log_{10} 2-\log_{10} 3^2
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\log_{10} 2-2\log_{10} 3
.\end{array}
Substituting the known values of the logarithmic expressions results to
\begin{array}{l}\require{cancel}
0.3010-2(0.4771)
\\\\=
0.3010-0.9542
\\\\=
-0.6532
.\end{array}