Answer
$x=\dfrac{9}{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\log_2(5x-6)-\log_2(x+1)=\log_23
,$ use the properties of logarithms to simplify the left-hand expression. Then drop the logarithm on both sides. Use the properties of equality and concepts on solving quadratic equations. Finally, do checking of the solutions with the original equation.
$\bf{\text{Solution Details:}}$
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
\log_2\dfrac{5x-6}{x+1}=\log_23
.\end{array}
Since the logarithm on both sides have the same base, then the logarithm can be dropped. Hence, the equation above is equivalent to \begin{array}{l}\require{cancel}
\dfrac{5x-6}{x+1}=3
.\end{array}
Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{5x-6}{x+1}=\dfrac{3}{1}
\\\\
(5x-6)(1)=(x+1)(3)
\\\\
5x-6=3x+3
\\\\
5x-3x=3+6
\\\\
2x=9
\\\\
x=\dfrac{9}{2}
.\end{array}
Upon checking, $
x=\dfrac{9}{2}
$ satisfies the original equation.