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:

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

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

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

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

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

$\small\longrightarrow \sf{log_{10}( \dfrac{3x}{9} ) = 0}$

$\small\sf{ \Longrightarrow \dfrac{3x}{9} = {10}^{0} }$

$\small\longrightarrow \sf{ \dfrac{x}{3} = 1}$

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

Next $\downarrow$

$\small\longrightarrow \sf{{x}^{2} - 1 = 0}$

$\small\longrightarrow \sf{ {x}^{2} = 1}$

$\small\longrightarrow \sf{x = \pm \sqrt{1} }$

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

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

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

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