# How do you solve the equation log_3(x+5)-log_3(x-7)=2?

Feb 17, 2015

You can use the following facts:

${\log}_{a} M - {\log}_{a} N = \log \left(\frac{M}{N}\right)$ and:

${\log}_{a} b = x \implies {a}^{x} = b$

So you get:

${\log}_{3} \left(x + 5\right) - {\log}_{3} \left(x - 7\right) = 2$
${\log}_{3} \left(\frac{x + 5}{x - 7}\right) = 2$
$\frac{x + 5}{x - 7} = {3}^{2}$
$x + 5 = 9 \left(x - 7\right)$
$x - 9 x = - 63 - 5$
$- 8 x = - 68$
$x = \frac{68}{8} = \frac{17}{2}$