#### Answer

$x=-3$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x=\log_6 \dfrac{1}{216}
,$ use the laws of exponents and the properties of logarithms to find an equivalent expression for the logarithmic expression.
$\bf{\text{Solution Details:}}$
Using exponents, the value $\dfrac{1}{216}$ is equivalent to $\dfrac{1}{6^3}.$ Hence, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\log_6 \dfrac{1}{6^3}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\log_6 6^{-3}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=-3\log_6 6
.\end{array}
Since $\log_b b=1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=-3(1)
\\\\
x=-3
.\end{array}