Answer
Since $x \ne - 1$, then $x = 3$
Work Step by Step
$\log_2(x-1) - \log_2 (x+3) = \log_2 (\frac{1}{x})$
$\log_2 \frac{x-1}{x+3} = \log_2 (\frac{1}{x})$
$\frac{x-1}{x+3} = \frac{1}{x}$
$x-1 = \frac{x+3}{x}$
$x(x-1) = x+3$
$x^{2} - x = x+3$
$x^{2} - x - x - 3 = 0$
$x^{2} - 2x - 3 = 0$
$(x-3)(x+1) = 0$
$x = 3, -1$
Since $x \ne - 1$, then $x = 3$
