# How do you solve log_3 x-log_3(x+5)=3?

Sep 11, 2016

$x = - \frac{135}{26} = 5 \frac{5}{26}$

#### Explanation:

"If the log terms are being subtracted, then the numbers are being divided"

${\log}_{3} \left(\frac{x}{x + 5}\right) = 3$

"Both sides must be logs, or both sides must be numbers"

${\log}_{3} 3 = 1 \text{ } \rightarrow 3 {\log}_{3} 3 = 3 \times 1 = 3$

${\log}_{3} \left(\frac{x}{x + 5}\right) = 3 {\log}_{3} 3$

${\log}_{3} \left(\frac{x}{x + 5}\right) = {\log}_{3} {3}^{3}$

$\therefore \frac{x}{x + 5} = \frac{27}{1} \text{ } \left({3}^{3} = 27\right)$

$27 \left(x + 5\right) = x$

$27 x + 135 = x$

$26 x = - 135$

$x = - \frac{135}{26}$