Question

Use the properties of logarithms to expand the following expression as much as possible.

Answer 1

`\log_2(2x^2+8x+8)`

1. Factor the expression in parenthesis:

`\begin{gathered} 2x^2+8x+8 \\ \\ =2(x^2+4x+4) \\ =2(x+2)\placeholder{⬚}^2 \end{gathered}`

2. Rewrite the expression using the factors above:

`\log_2(2(x+2)\placeholder{⬚}^2)`

3. Use the next properties to expand the expression:

`\begin{gathered} \log_a(x*y)=\log_ax+\log_ay \\ \log_aa=(\log_a)/(\log_a)=1 \\ \log_a(b^c)=c*\log_ab \end{gathered}`
`\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}`

Then, the given expression is equal to:

`\log_2(2x^2+8x+8)=1+2\log_2(x+2)`