Answer
$x=-\dfrac{42}{23}$
Work Step by Step
Using properties of logarithms, then,
\begin{array}{l}
-\log_6(4x+7)+\log_6x=1\\\\
\log_6x-\log_6(4x+7)=1\\\\
\log_6\left( \dfrac{x}{4x+7} \right)=1\\\\
\dfrac{x}{4x+7}=6^1\\\\
x=6(4x+7)\\\\
x=24x+42\\\\
-23x=42\\\\
x=-\dfrac{42}{23}
.\end{array}
