[tex]\log_2(2x^2+8x+8)[/tex]
1. Factor the expression in parenthesis:
[tex]\begin{gathered} 2x^2+8x+8 \\ \\ =2(x^2+4x+4) \\ =2(x+2)\placeholder{⬚}^2 \end{gathered}[/tex]
2. Rewrite the expression using the factors above:
[tex]\log_2(2(x+2)\placeholder{⬚}^2)[/tex]
3. Use the next properties to expand the expression:
[tex]\begin{gathered} \log_a(x*y)=\log_ax+\log_ay \\ \log_aa=\frac{\log_a}{\log_a}=1 \\ \log_a(b^c)=c*\log_ab \end{gathered}[/tex][tex]\begin{gathered} \log_2(2(x+2)\placeholder{⬚}^2) \\ =\log_22+\log_2((x+2)\placeholder{⬚}^2) \\ =1+2\log_2(x+2) \end{gathered}[/tex]
Then, the given expression is equal to:
[tex]\log_2(2x^2+8x+8)=1+2\log_2(x+2)[/tex]
