Answer
$\left\{\dfrac{1}{8}\right\}$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_3(x+1)-\log_3x=2
,$ is equivalent to
\begin{align*}\require{cancel}
\log_3\dfrac{x+1}{x}&=2
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
.\end{align*}
Since $\log_b y=x$ implies $y=b^x$, the equation above implies
\begin{align*}\require{cancel}
\dfrac{x+1}{x}&=3^2
\\\\
\dfrac{x+1}{x}&=9
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\cancel x\cdot\dfrac{x+1}{\cancel x}&=9\cdot x
\\\\
x+1&=9x
\\
1&=9x-x
\\
1&=8x
\\\\
\dfrac{1}{8}&=\dfrac{\cancel{8}x}{\cancel{8}}
\\\\
\dfrac{1}{8}&=x
.\end{align*}
Hence, the solution set of the equation $
\log_3(x+1)-\log_3x=2
$ is $
\left\{\dfrac{1}{8}\right\}
$.
