[tex]\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}[/tex]
Given:
▪ [tex]\sf{f(x) = ( log(3x) - 2log(3))( {x}^{2} - 1) }[/tex] [tex]\ \sf{x \in R}[/tex]
Solve:
[tex]\longrightarrow \sf{f(x) = (log(3x) - 2log(3))( {x}^{2} - 1) = 0}[/tex]
[tex]\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0 \: \: or \: {x}^{2} - 1 = 0 }[/tex]
[tex]\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0}[/tex]
[tex]\small\longrightarrow \sf{ log(3x) - 2 log(3) = 0}[/tex]
[tex]\small\longrightarrow \sf{ log(3x) - log( {3}^{2} ) = 0}[/tex]
[tex]\small\longrightarrow \sf{log_{10}( \dfrac{3x}{9} ) = 0}[/tex]
[tex] \small\sf{ \Longrightarrow \dfrac{3x}{9} = {10}^{0} }[/tex]
[tex]\small\longrightarrow \sf{ \dfrac{x}{3} = 1}[/tex]
[tex]\small \sf{x= \underline{3}}[/tex]
Next [tex]\downarrow[/tex]
[tex]\small\longrightarrow \sf{{x}^{2} - 1 = 0}[/tex]
[tex]\small\longrightarrow \sf{ {x}^{2} = 1}[/tex]
[tex]\small\longrightarrow \sf{x = \pm \sqrt{1} }[/tex]
[tex]\small\sf{x= (\underline{1},-1)}[/tex]
[tex]\small\longrightarrow \sf{ log(3x) = log(3( - 1)) \cancel{ \in}R}[/tex]
[tex]\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}[/tex]
[tex]\small\bm{The \: \: roots \: \: of \: \: f(x) \: are \: 1 \: and \: 3.}[/tex]
