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

Jul 21, 2016

$x = \frac{1 + \sqrt{1 + 4 e}}{2} = 2.22287$, nearly.

#### Explanation:

$x > 1$ to make both $\ln x \mathmr{and} \ln \left(x - 1\right)$ real.

ln x + ln ( x - 1 ) = ln (x(x-1)) = 1 = ln e#.

So, $x \left(x - 1\right) = e$. Solving,

$x . = . \frac{1 \pm \sqrt{1 + 4 e}}{2}$

Negative root is inadmissible.

The positive root is 2,22287, nearly .