Answer
$x \approx 7.266$
Work Step by Step
$\log_3 (x-4) + \log_3 (x+1) = 3$
$\log_3 (x-4)(x+1) = 3$
$(x-4)(x+1) = 3^{3}$
$x(x+1)-4(x+1) = 27$
$x^{2} + x - 4x - 4 = 27$
$x^{2} - 3x - 31 = 0$
$x = \frac{-b±\sqrt {b^{2}-4ac}}{2a}$
$x = \frac{-(-3)±\sqrt {(-3)^{2}-4(1)(-31)}}{2(1)}$
$x = \frac{3±\sqrt {9-4(1)(-31)}}{2}$
$x = \frac{3±\sqrt {9+124}}{2}$
$x = \frac{3±\sqrt {133}}{2}$
$x = -4.266..., 7.266...$
($x$ must be greater than 4.)
$x \approx 7.266$
Check:
$\log_3 (x-4) + \log_3 (x+1) \overset{?}{=} 3$
$\log_3 ((7.266...)-4) + \log_3 ((7.266...)+1) \overset{?}{=} 3$
$\log_3 (3.266...) + \log_3 (8.266...) \overset{?}{=} 3$
$\log_3 (3.266...)(8.266...) \overset{?}{=} 3$
$\log_3 (27)\overset{?}{=} 3$
$\log_3 (3^{3})\overset{?}{=} 3$
$3\log_3 (3)\overset{?}{=} 3$
$3(1)\overset{?}{=} 3$
$3 = 3$
