Answer
$\left\{-6+\sqrt[3]{25}\right\}$
Work Step by Step
Since $\log_b y=x$ implies $y=b^x$, the given equation, $
\log_5(x+6)^3=2
$, implies
\begin{align*}\require{cancel}
(x+6)^3&=5^2
\\
(x+6)^3&=25
.\end{align*}
Taking the cube root of both sides, the equation above is equivalent to
\begin{align*}\require{cancel}
x+6&=\sqrt[3]{25}
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x+6-6&=-6+\sqrt[3]{25}
\\
x&=-6+\sqrt[3]{25}
.\end{align*}
Hence, the solution set of the equation $
\log_5(x+6)^3=2
$ is $
\left\{-6+\sqrt[3]{25}\right\}
$.
