Answer
$x= \frac{-1+\sqrt{9+4e}}{2}\approx 1.7290$
Work Step by Step
$\ln(x-1)+\ln(x+2)=1$
$\ln[(x-1)(x+2)]=1$
$(x-1)(x+2)=e^1$
$x^{2}+x-2=e^1$
$x^{2}+x-2-e^1=0$
$x^{2}+x-(2+e)=0$
We use the quadratic formula:
$x= \frac{-1\pm\sqrt{1^2-4(1)(-2-e)}}{2*1}=\frac{-1\pm\sqrt{9+4e}}{2}$
However, $x= \frac{-1-\sqrt{9+4e}}{2}$ is negative and produces a negative log in the original equation (undefined). So we throw this solution out.
Therefore, the only solution is:
$x= \frac{-1+\sqrt{9+4e}}{2}\approx 1.7290$
