Answer
$x = \frac{5+\sqrt {481}}{2} \approx 13.466$
Work Step by Step
$\log (x+2) + \log(x-7) = 2$
$\log(x+2)(x-7) = 2$
$\log_{10} (x+2)(x-7) = 2$
$10^{2} = (x+2)(x-7)$
$100 = (x+2)(x-7)$
$x(x-7) +2 (x-7) = 100$
$x^{2} - 7x + 2x - 14 = 100$
$x^{2} - 5x - 14 - 100 = 0$
$x^{2} - 5x - 114 = 0$
$x = \frac{-(-5)±\sqrt {(-5)^{2}-4(1)(-114)}}{2(1)}$
$x = \frac{5±\sqrt {25-4(1)(-114)}}{2}$
$x = \frac{5±\sqrt {25+456}}{2}$
$x = \frac{5±\sqrt {481}}{2}$
Since we can't take the log of a negative number, the only possible solution is $x = \frac{5+\sqrt {481}}{2}$.
$x = \frac{5+\sqrt {481}}{2} \approx 13.466$
Check:
$\log ((13.466)+2) + \log(6.465...) \overset{?}{=}2$
$\log (15.4658...) + \log(6.4658...) \overset{?}{=} 2$
$\log (15.4658...)(6.4658...) \overset{?}{=} 2$
$\log 100 \overset{?}{=} 2$
$\log_{10} 10^{2} \overset{?}{=} 2$
$2(\log_{10} 10) \overset{?}{=} 2$
$2(1) \overset{?}{=} 2$
$2 = 2$
