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

Jul 9, 2018

$\textcolor{b l u e}{x = \frac{- {e}^{2}}{1 - {e}^{2}} \approx 1.156517643}$

#### Explanation:

$\ln a - \ln b = \ln \left(\frac{a}{b}\right)$

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

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

${e}^{\ln \left(\frac{x}{x - 1}\right)} = {e}^{2}$

$\frac{x}{x - 1} = {e}^{2}$

$x - {e}^{2} x = - {e}^{2}$

$x \left(1 - {e}^{2}\right) = - {e}^{2}$

$x = \frac{- {e}^{2}}{1 - {e}^{2}} \approx 1.156517643$