Help with a logarithmic function

Tanamanis

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Dec 1, 2019
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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

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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

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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

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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

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\(\displaystyle {R} \ \setminus \{-2 \le x \le 2 \} \ \) 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

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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

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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: \(\displaystyle (-\infty,-2)\cup(2,\infty)\) .
 
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