Answer
Solution set = $\{5\}.$
Work Step by Step
For the equation to be defined, logarithms must have positive arguments
$\left[\begin{array}{llll}
5x+1 \gt 0, & and & 2x+3 \gt 0 & \\
x \gt -1/5 & & x\gt-3/2 &
\end{array}\right]$
$ x \gt -1/5 \qquad (*)$
This is the condition eventual solutions must satisfy.
$\log(5x+1)=\log(2x+3)+\log 2\qquad $... RHS: product rule
$\log(5x+1)=\log[(4x+6)\cdot 2]\quad $ ... $\log $ is one-to-one
$5x+1=4x+6$
$5x-4x=6-1$
$ x=5\qquad $ ... satisfies (*)
Thus, the solution set = $\{5\}.$
