## College Algebra (11th Edition)

$x=-3$
$\bf{\text{Solution Outline:}}$ 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}