# How do you solve log_3(x+16) - log_3(x) = log_3(2)?

Jun 7, 2016

$x = 16$

#### Explanation:

First, note that they are all in ${\log}_{3}$, so we simplify the left-hand side to get

${\log}_{3} \left(\frac{x + 16}{x}\right) = {\log}_{3} 2$
$\implies \frac{x + 16}{x} = 2$, $2 x = x + 16$, $x = 16$

Jun 7, 2016

$16 = x$

#### Explanation:

The log laws suggest that when to log's with the same base subtract from one another, it can be written as

${\log}_{a} \left(x\right) - {\log}_{a} \left(y\right) = {\log}_{a} \left(\frac{x}{y}\right)$

Therefore, our equation can be simplified into

${\log}_{3} \left(\frac{x + 16}{x}\right) = {\log}_{3} \left(2\right)$

Therefore $\frac{x + 16}{x} = 2$

$x + 16 = 2 x$

$16 = x$