Question

What are the conditions under which the logarithm ${\log }_{2x}\left(x^2-1\right)$ is defined ?

• A. x > 0
• B. x > 1
• C. x > 1 and x < -1
• D. x < -1

Answer 1

The correct option is B

x > 1

We learn that for a logarithm ${\log }_{b}a$ to be defined, the following must be satisfied.
$a>0b>0;b\ne 1$

Given log expression = ${\log }_{2x}\left(x^2-1\right)$$2x>0;2x\ne 1$
$x>0;x\ne \frac{1}{2}$ ...(1)

Also $x^2-1>0$
$(x-1)(x+1)>0.$
This is true when x > 1 or x < -1 ...(2)

Combining (1) and (2) we can say x > 1 is the required condition.