Answer
$x = 2$
Work Step by Step
$\ln (x+4) - \ln (x+1) = \ln x$
$\ln \frac{x+4}{x+1} = \ln x$
$\frac{x+4}{x+1} = x$
$x+4 = x(x+1)$
$x+4 = x^{2} + x$
$x^{2} + x - x - 4 = 0$
$x^{2} - 4 = 0$
$x^{2} = 4$
$x = ±2$
Since $x\ne -2$, then $x = 2$