Domain of One Variable Function – A function with ln inside a fraction – Exercise 5478

Exercise

Determine the domain of the function:

$f(x)=\frac{\ln (x^2)}{\ln^2 (x) -4}$

Final Answer

Show final answer

Solution

Let’s find the domain of the function:

$f(x)=\frac{\ln (x^2)}{\ln^2 (x) -4}$

There is a ln function, so we need the expressions inside the ln to be positive:

$x^2>0\text{  and  }x> 0$

The two inequalities result

$x>0$

There is also a denominator, therefore the denominator must be different from zero. We check when it equals zero:

$\ln^2 (x) -4= 0$

$\ln^2 (x)= 4$

$\ln (x)= \pm 2$

We got two solutions. One solution,

$\ln x=2$

We use the logarithm definition and get

$x=e^2$

Second solution,

$\ln x=-2$

Again, we use the logarithm definition and get

$x=e^{-2}$

Since we require the denominator to be other than zero, we get that x cannot hold these values. That is,

$x\neq e^2 , e^{-2}$

In summary, the function domain is

$(0,e^{-2}) \cup (e^{-2},e^2) \cup (e^2,\infty)$

Note: The meaning of the sign:

$\cup$

is union (“or” relation).

Have a question? Found a mistake? – Write a comment below!
Was it helpful? You can buy me a cup of coffee here, which will make me very happy and will help me upload more solutions!