Simple question about the natural logarithm

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January 16th 2009, 09:57 AM
Kaitosan

Simple question about the natural logarithm

Am I right to assume that WHENEVER an expression containing a natural logarithm is produced from integrating an expression, the expression always contain the absolute value symbol? The reason I'm asking this is because my book sometimes confuse me by alternating between using the absolute value symbol and forgoing it when producing a natural logarithmic expression from integration.

Also, why do we even use the absolute value symbol for natural logarithms? We know that ln[abs(x)] = ln x.
January 16th 2009, 10:10 AM
Danny

Quote:

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January 16th 2009, 10:29 AM
Plato

Quote:

Originally Posted by Kaitosan

Also, why do we even use the absolute value symbol for natural logarithms? We know that ln[abs(x)] = ln x.

Do we know that?
The domain of $f(x) = \ln \left( {\left| x \right|} \right)$ is $\mathbb{R}\backslash \left\{ 0 \right\}$ .
While the domain of $g(x) = \ln \left( x \right)$ is $\left( {0,\infty } \right)$ .
January 16th 2009, 10:55 AM
Kaitosan

I'm sorry, but I don't quite understand that domain thingy. :(

I'll give an example from my book -

INTEG: 2x/(4+9x^2) dx

Answer: (1/9) ln (4+9x^2) + c (no absolute value)

Why doesn't it have the absolute value symbol?
January 16th 2009, 10:59 AM
Plato

Quote:

Originally Posted by Kaitosan

I'll give an example from my book -
INTEG: 2x/(4+9x^2) dx
Answer: (1/9) ln (4+9x^2) + c (no absolute value)
Why doesn't it have the absolute value symbol?

Because $4+9x^2$ is always positive.
There can be no domain issues there.
January 16th 2009, 11:58 AM
Kaitosan

Oooh ok thanks! I don't know why I didn't realize it before lol.
January 16th 2009, 01:04 PM
Kaitosan

I've another question. Let's use the following example -

INTEG: (4x^2 - 2x + 2)/[(x^2 + 1)(x-1)] dx

After using partial fractions, the answer is, according to my book, ln[abs(x^2 + 1)] + 2 ln [abs(x-1)] + C

Now, obviously, the first ln expression cannot be negative so why is there an absolute value symbol in it? Is it because of the other ln expression? I'd appreciate some clarification. Thanks!
January 16th 2009, 01:25 PM
Plato

Quote:

Originally Posted by Kaitosan

INTEG: (4x^2 - 2x + 2)/[(x^2 + 1)(x-1)] dx
After using partial fractions, the answer is, according to my book, ln[abs(x^2 + 1)] + 2 ln [abs(x-1)] + C
Now, obviously, the first ln expression cannot be negative so why is there an absolute value symbol in it? Is it because of the other ln expression? I'd appreciate some clarification.

What harm is done by using them there?
The fact we don't need them does not change the answer in an way.
January 16th 2009, 01:35 PM
Kaitosan

lol.................... ok. Thanks. (Rock)