# How do you solve ln(x+4)-ln(x+3)=lnx?

Mar 23, 2016

$x = - 1 + \sqrt{5}$

#### Explanation:

Dominion: color(blue)(x+4>0 and x+3>0 and x>0 <=> x>0

$\ln \left(x + 4\right) - \ln \left(x + 3\right) = \ln \left(x\right)$

$\ln \left(x + 4\right) = \ln \left(x + 3\right) + \ln \left(x\right)$

The sum of logarithms is the logarithm of the product

$\ln \left(x + 4\right) = \ln \left(\left(x + 3\right) x\right)$

color(blue)(ln(a)=ln(b) ->a=b

$x + 4 = \left(x + 3\right) x$

$x + 4 = {x}^{2} + 3 x$

${x}^{2} + 2 x - 4 = 0$

$x = \frac{- 2 \pm \sqrt{{2}^{2} - 4 \cdot 1 \cdot \left(- 4\right)}}{2}$

$x = \frac{- 2 \pm \sqrt{4 + 16}}{2}$

$x = \frac{- 2 \pm \sqrt{20}}{2}$

$x = \frac{- 2 \pm 2 \sqrt{5}}{2}$

$x = - 1 \pm \sqrt{5}$

$x = - 1 + \sqrt{5}$, because x must be larger than 0