Help with a logarithmic function

[Tanamanis]

I really need help with this problem. I need to state the domain of the function f(x)= log(x^2-4), and I know the domain is all numbers except those between -2 and 2, but I don't understand why. Please help!

 

[MarkFL]

Hello, and welcome to FMH!

In order for $f(x)$ to return a real number, we need the argument for the log function to be positive, and so we must solve:

[MATH]x^2-4>0[/MATH]
What do you find?

 

[pka]

I really need help with this problem. I need to state the domain of the function f(x)= log(x^2-4), and I know the domain is all numbers except those between -2 and 2, but I don't understand why.

Here is a pet peeve of mine( as well as the tradition I stand). The phrase between -2 and 2 means the set $\displaystyle (-2,2)=\{x: -2<x<2\}$ that an open interval. If we want the closed interval $\displaystyle [-2,2]=\{x: -2\le x\le 2\}$, the phrase is from -2 to 2.
What this has to do with your question is the fact the logarithm function is defined positive real numbers (we are not considering complex numbers).
That is why the function $\displaystyle f(x)=\log(x^2-4)$ must exclude the closed interval $\displaystyle [-2,2]$ because that function is only defined for positive real numbers. In set notation the domain is written as $\displaystyle \mathbb{R}\setminus [-2,2]$.
I hope you understand that $\displaystyle x\ne\pm 2$ because $\displaystyle (\pm 2)^2-4=0$ but $\displaystyle \log(0)$ doex not exist.
If you do have more question, please do ask but with detailed questions.

 

[lookagain]

Post # 3 mentions set notation for the domain, but it contains interval notation
in it.

$\displaystyle {R} \ \setminus \{-2 \le x \le 2 \} \$ could be one of the ways for set notation for the domain.

Or, $\displaystyle \ \{x < -2 \} \ \cup \ \{x > 2 \} \ \$ could be another.

 

[topsquark]

$\displaystyle {R} \ \setminus \{-2 \le x \le 2 \} \$ could be one of the ways for set notation for the domain.

Actually that should be $$\mathbb{R} \setminus \{ x \in \mathbb{R} | -2 \leq x \leq 2 \}$$ ? (Set notation can be such a pain!)

-Dan

Addendum: Okay, maybe the $$x \in \mathbb{R}$$ is overkill since the statement (sort of) implies that the set being considered is $$\mathbb{R}$$ already.

 

[Steven G]

I really need help with this problem. I need to state the domain of the function f(x)= log(x^2-4), and I know the domain is all numbers except those between -2 and 2, but I don't understand why. Please help!

You can only compute the log of positive values. You happen to be taking the log of x^2-4 so it is x^2-4 that has to be positive. You should know how to graph this. So do so and see for which x, x's, if any that makes x^2-4 >0. That will be your domain.

The real question this problem is asking is for you to solve x^2-4>0

 

[pka]

Post # 3 mentions set notation for the domain, but it contains interval notation
in it. $\displaystyle {R} \ \setminus \{-2 \le x \le 2 \} \$ could be one of the ways for set notation for the domain.
Or, $\displaystyle \ \{x < -2 \} \ \cup \ \{x > 2 \} \ \$ could be another.

Why not cut to the chase? The domain is: $\displaystyle (-\infty,-2)\cup(2,\infty)$ .