# Help with a logarithmic function

#### Tanamanis

##### New member
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

##### Super Moderator
Staff member
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:

$$\displaystyle x^2-4>0$$

What do you find?

#### pka

##### Elite Member
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

##### Elite Member
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.

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#### topsquark

##### Full Member
$$\displaystyle {R} \ \setminus \{-2 \le x \le 2 \} \$$ could be one of the ways for set notation for the domain.
Actually that should be $$\displaystyle \mathbb{R} \setminus \{ x \in \mathbb{R} | -2 \leq x \leq 2 \}$$ (Set notation can be such a pain!)

-Dan

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

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#### Jomo

##### Elite Member
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

##### Elite Member
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: $$\displaystyle (-\infty,-2)\cup(2,\infty)$$ .