Question

Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluatedwithout a calculator,log2 (8x2 + 96x + 288)

Answer 1

Given:

`\log _2(8x^2+96x+288)`

Simplify the expression,

`\begin{gathered} \log _2(8x^2+96x+288) \\ \text{First take}(8x^2+96x+288) \\ 8x^2+96x+288=8(x^2+12x+36) \\ =8(x^2+6x+6x+36) \\ =8(x(x+6)+6(x+6)) \\ =8(x+6)(x+6) \\ =8(x+6)^2 \end{gathered}`

Now, simplify the logarithmic expression,

`\begin{gathered} \log _2(8x^2+96x+288)=\log _28(x+6)^2 \\ \text{Apply the rule: }\log _a(bc)=\log _ab+\log _ac \\ \log _28(x+6)^2=\log _28+\log _2(x+6)^2 \\ \text{Apply the rule}\colon\log _a(x^b)=b\log _ax \\ \log _28+\log _2(x+6)^2=\log _22^3+\log _2(x+6)^2 \\ =3\log _22+2\log _2(x+6) \\ \text{Apply the rule}\colon\log _aa=1 \\ =3+2\log _2(x+6) \end{gathered}`

Answer:

`3+2\log _2(x+6)`