+26396 Help with Log Question!
Solve for x.
\(\begin{array}{|rcll|} \hline \log_2(x-2)+\log_2(x+6) &=& 7 \quad & | \quad \log(a*b) = \log(a)+ \log(b) \\ \log_2(~(x-2)*(x+6)~) &=& 7 \quad & | \quad 2^{()} \\ (x-2)*(x+6) &=& 2^7 \\ (x-2)*(x+6) &=& 128 \\ (x-2)*(x+6) -128 &=& 0 \\ x^2+6x-2x-12-128 &=& 0 \\ x^2+4x -140 &=& 0 \\\\ x &=& \dfrac{-4\pm \sqrt{16-4*(-140)} }{2} \\ x &=& \dfrac{-4\pm \sqrt{16+4*140} }{2} \\ x &=& \dfrac{-4\pm \sqrt{576} }{2} \\ x &=& \dfrac{-4\pm 24 }{2} \\ x &=& -2\pm 12 \\\\ x_1 &=& -2+ 12 \\ \mathbf{x_1} & \mathbf{=}& \mathbf{10} \\\\ x_2 &=& -2- 12 \\ \mathbf{x_2} & \mathbf{=}& \mathbf{-14} \\ && x_2 \text{ is no solution,}\\ && \text{because } x_2 < 0 ! \\ \hline \end{array}\)
