Answer
$x=- 4 +\sqrt{17}\approx 0.123$
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_2[(x+1)(x+7)] $
Since $\log_m{n} = 1 $ gives: $m^{(1)}=n$, then we have:
$\log_2(x^2+8x+7)=3$
$x^2+8x+7=2^3$
or, $x^2+8x-1=0$
This is a quadratic equation; thus by using the quadratic formula $x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$, it will become:
$x=\dfrac{-8 \pm \sqrt{8^2-(4)(1)(1)}}{2(1)}$
or, $x=-4 + \sqrt{17}, - 4 - \sqrt{17}$
Since, the domain of the variable is $x \gt 0$, we cannot accept the value of $x=- 4 - \sqrt{17}$
Thus, our answer is: $x=- 4 +\sqrt{17}\approx 0.123$
