Question

What are the roots of the function f(x) = (log(3x ) − 2 log(3)) · (x 2 − 1) with x ∈ R?

Answer 1

`\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}`

Given:

▪
`\sf{f(x) = ( log(3x) - 2log(3))( {x}^(2) - 1) }`
`\ \sf{x \in R}`

Solve:

`\longrightarrow \sf{f(x) = (log(3x) - 2log(3))( {x}^(2) - 1) = 0}`

`\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0 \: \: or \: {x}^(2) - 1 = 0 }`

`\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0}`

`\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0}`

`\small\longrightarrow \sf{ log(3x) - log( {3}^(2) ) = 0}`

`\small\longrightarrow \sf{log_(10)( (3x)/(9) ) = 0}`

`\small\sf{ \Longrightarrow (3x)/(9) = {10}^(0) }`

`\small\longrightarrow \sf{ (x)/(3) = 1}`

`\small \sf{x= \underline{3}}`

Next
`\downarrow`

`\small\longrightarrow \sf{{x}^(2) - 1 = 0}`

`\small\longrightarrow \sf{ {x}^(2) = 1}`

`\small\longrightarrow \sf{x = \pm √(1) }`

`\small\sf{x= (\underline{1},-1)}`

`\small\longrightarrow \sf{ log(3x) = log(3( - 1)) \cancel{ \in}R}`

`\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}`

`\small\bm{The \: \: roots \: \: of \: \: f(x) \: are \: 1 \: and \: 3.}`