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

Jun 20, 2016

$x = \frac{1}{2} \left(- 1 + \sqrt{9 + 4 e}\right) = 1.72896$

Explanation:

$L n \left(x - 1\right) + \ln \left(x + 2\right) = 1 \to L n \left(x - 1\right) \left(x + 2\right) = L n e$

$\left(x - 1\right) \left(x + 2\right) - e = 0$

Solving for $x$

$x = \frac{1}{2} \left(- 1 \pm \sqrt{9 + 4 e}\right) = \left\{- 2.72896 , 1.72896\right\}$

Now, substituting those values in $\left(x - 1\right)$ and $\left(x + 2\right)$ keeping in mind that both must be positive, we get at the solution.

$x = \frac{1}{2} \left(- 1 + \sqrt{9 + 4 e}\right) = 1.72896$