How do you solve ln(x+1)-ln(x-2) = ln x?

Apr 3, 2018

$x = 3.30$

Explanation:

$I n \left(x + 1\right) - I n \left(x - 2\right) = I n x$
$I n \frac{x + 1}{x - 2} = I n x$
$\frac{x + 1}{x - 2} = x$
$x + 1 = x \left(x - 2\right)$
$x + 1 = {x}^{2} - 2 x$
${x}^{2} - 3 x - 1 = 0$

$x = \frac{3 \pm \sqrt{9 + 4}}{2}$
$x = \frac{3 + \sqrt{13}}{2} \mathmr{and} \frac{3 - \sqrt{13}}{2}$
$x = 3.30 \mathmr{and} - 0.302$
BUT you can't log a NEGATIVE number so $x = - 0.302$
Therefore, $x = 3.30$ is the only answer