Answer
$\left\{2\right\}$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_3(x+2)-\log_3 x=\log_3 2
$, is equivalent to
\begin{align*}\require{cancel}
\log_3\dfrac{x+2}{x}&=\log_3 2
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
.\end{align*}
Since $\log_b x=\log_b y$ implies $x=y$, the equation above implies
\begin{align*}\require{cancel}
\dfrac{x+2}{x}&=2
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\cancel x\cdot\dfrac{x+2}{\cancel x}&=2\cdot x
\\\\
x+2&=2x
\\
x-x+2&=2x-x
\\
2&=x
.\end{align*}
Hence, the solution set of the equation $
\log_3(x+2)-\log_3 x=\log_3 2
$ is $
\left\{2\right\}
$.
