# How do you solve log_3 x-log_3(x-1)=1?

Aug 17, 2016

$x = \frac{3}{2}$

#### Explanation:

${\log}_{3} x - {\log}_{3} \left(x - 1\right) = 1 = {\log}_{3} 3$ so
${\log}_{3} \left(\frac{x}{x - 1}\right) = {\log}_{3} 3$ so
$\frac{x}{x - 1} = 3$ or
$x = 3 \left(x - 1\right)$ Solving for $x$ gives

$x = \frac{3}{2}$. Verifying the feasibility

${\log}_{3} \left(\frac{3}{2}\right) - {\log}_{3} \left(\frac{1}{2}\right) = {\log}_{3} 3$

So, $x = \frac{3}{2}$ is the solution.