Answer
$x=\dfrac{-5+3 \sqrt{5}}{2} \approx 0.854$
Work Step by Step
Apply the logarithmic property : $\log_a M+\log_a N = \log_a (MN)$ and rearrange the given expression to obtain:
$\log_3[(x+1)(x+4)] $
Since, $\log_m{n} = 1 $ gives: $m^{(1)}=n$, then we have:
$\log_3(x^2+5x+4)=2$
$x^2+5x+4=3^2$
or, $x^2+5x-5=0$
This is a quadratic equation; thus by using the quadratic formula $x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$, we get:
$x=\dfrac{-5 \pm 3 \sqrt{5}}{2(1)}$
or, $x=\dfrac{-5 + 3 \sqrt{5}}{2}, \dfrac{-5 -3 \sqrt{5}}{2}$
Since, the domain of the variable is $x \gt 0$, we cannot accept the value of $x=\dfrac{-5 -3 \sqrt{5}}{2}$
Thus, our answer is: $x=\dfrac{-5+3 \sqrt{5}}{2} \approx 0.854$
