I was assigned the following:
The only solution of the equation ln(x) + ln(x-2) =1 is x = ?
I am unsure how to solve for x.
Use the fact that $\displaystyle \ln(a)+\ln(b)=\ln(a\cdot b)$ and $\displaystyle \ln(e)=1$ therefore the equation can be arranged as:
$\displaystyle \ln[x(x-2)]=\ln(e)$
$\displaystyle \Leftrightarrow x(x-2)=e$
Solve this quadratic equation.
Following on from Siron's working:
$\displaystyle x^2-2x = e \Leftrightarrow x^2-2x-e = 0$
Since e is a number solve using the quadratic formula:
$\displaystyle x = \dfrac{2\pm \sqrt{4+4e}}{2} = \dfrac{2 \pm \sqrt{4(1+e)}}{2} = \dfrac{2 \pm \sqrt{4}\sqrt{1+e}}{2}$
$\displaystyle = \dfrac{2 \pm 2\sqrt{1+e}}{2} = \dfrac{2(1 \pm \sqrt{1+e})}{2} =1 \pm \sqrt{1+e}$
Since $\displaystyle \sqrt{1+e} > 1$ discard the negative solution due to domain issues (x>2)
$\displaystyle x = 1 + \sqrt{1+e}$