Exercise
Determine the domain of the function:
f(x)=\frac{\ln (x^2)}{\ln^2 (x) -4}
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).
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