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