# How do you solve  lnx + ln(x+1) = ln12?

Oct 23, 2015

$x = 3$

#### Explanation:

In general
$\textcolor{w h i t e}{\text{XXX}} \log \left(A\right) + \log \left(B\right) = \log \left(A B\right)$
(this is one of the basic logarithmic rules)

Specifically
$\textcolor{w h i t e}{\text{XXX}} \ln \left(x\right) + \ln \left(x + 1\right) = \ln \left({x}^{2} + x\right)$
and we are told, this is
$\textcolor{w h i t e}{\text{XXXXXXXXXXXXX}} = \ln \left(12\right)$

$\Rightarrow {x}^{2} + x = 12$

$\textcolor{w h i t e}{\text{XXX}} {x}^{2} + x - 12 = 0$

$\textcolor{w h i t e}{\text{XXX}} \left(x + 4\right) \left(x - 3\right) = 0$

$\textcolor{w h i t e}{\text{XXX}} x = - 4$$\textcolor{w h i t e}{\text{XXX}}$or$\textcolor{w h i t e}{\text{XXX}} x = 3$

Since $\ln \left(x\right)$ is not defined for negative values of $x$
$\Rightarrow x = 3$