# How do you solve Ln 12 - ln(x - 1) = ln(x-2)?

Aug 28, 2016

$x = 5$

#### Explanation:

$\ln 12 - \ln \left(x - 1\right) = \ln \left(x - 2\right)$

$\Leftrightarrow \ln \frac{12}{x - 1} = \ln \left(x - 2\right)$ or

$\frac{12}{x - 1} = \left(x - 2\right)$ or

$12 = \left(x - 1\right) \left(x - 2\right)$ or

$x \left(x - 2\right) - 1 \left(x - 2\right) = 12$ or

${x}^{2} - 2 x - x + 2 = 12$ or

${x}^{2} - 3 x - 10 = 0$ or

${x}^{2} - 5 x + 2 x - 10 = 0$ or

$x \left(x - 5\right) + 2 \left(x - 5\right) = 0$ or

$\left(x - 5\right) \left(x + 2\right) = 0$ or

$x = 5$ or $x = - 2$

But as $x = - 2$ is extraneous as we cannot have log of a negative number,

Answer is $x = 5$.