Question

What are the roots of the function f(x) = (log(3x) − 2log(3)) · (x2 − 1) with x ∈ R?​

Answer 1

x = 1, x = 3

Explanation:

Given function:

`f(x)=\left(\log(3x)-2 \log(3)\right) \cdot \left(x^2-1\right)`

The roots of the function are when f(x) = 0:

`\begin{aligned}f(x) & = 0\\ \implies \left(\log(3x)-2 \log(3)\right) \cdot \left(x^2-1\right) & = 0\\\\ \textsf{Therefore,} \:\:\:\log(3x)-2 \log(3) & = 0\\ \textsf{and }\:\:\:x^2-1 & = 0 \end{aligned}`

Case 1

`\begin{aligned}\left(\log(3x)-2 \log(3)\right) & = 0\\\log(3x) & = 2 \log(3)\\ \log(3x) & = \log (3)^2\\ \log(3x) & = \log (9)\\ 3x & = 9\\ x & = 3\end{aligned}`

Case 2

`\begin{aligned}\left x^2-1\right & = 0\\x^2 & = 1\\ x & = \pm 1\end{aligned}`

As logs of negative numbers cannot be taken, x = 1 only.

Therefore, the roots of the given function are: x = 1, x = 3