Question

The domain of the function $f\left(x\right)={\log }_{\left(2x-1\right)}\left(x-1\right)$ is

• A. $\left(1,\infty \right)$
• B. $\left(\frac{1}{2},\infty \right)$
• C. $\left(0,\infty \right)$
• D. None of these

Answer 1

The correct option is A

$\left(1,\infty \right)$

Explanation for correct Option(s):

Finding the domain of the function $f\left(x\right)$:

Given function, $f\left(x\right)={\log }_{\left(2x-1\right)}\left(x-1\right)$

We know that ${\log }_{a}x$ is defined as $a\ne 1$, $a>0$ and $x>0$.

Applying $a\ne 1$on the given function,

$$\begin{gathered} \Rightarrow 2x-1\ne 1 \\ \Rightarrow 2x\ne 2 \\ \therefore x\ne 1 \end{gathered}$$

Applying $a>0$ on the given function,

$$\begin{gathered} \Rightarrow 2x-1>0 \\ \Rightarrow 2x>1 \\ \therefore x>\frac{1}{2} \end{gathered}$$

Applying $x>0$ on the given function,

$$\begin{gathered} \Rightarrow x-1>0 \\ \Rightarrow x>1 \\ \therefore x>1 \end{gathered}$$

By comparing the above three, the domain of the given function would be $\left(1,\infty \right)$.

Therefore, the correct answer is Option (A).