Answer
$x=-1+\sqrt[3]{49}$
Work Step by Step
Since $\log_b y=x$ implies $y=b^x$, the given equation, $
\log_7(x+1)^3=2
$, implies
\begin{align*}\require{cancel}
(x+1)^3&=7^2
\\
(x+1)^3&=49
.\end{align*}
Taking the cube root of both sides, the equation above is equivalent to
\begin{align*}\require{cancel}
x+1&=\sqrt[3]{49}
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
x&=-1+\sqrt[3]{49}
.\end{align*}
Hence, the solution to the equation $
\log_7(x+1)^3=2
$ is $
x=-1+\sqrt[3]{49}
$.