Exploring the Relationship between -ln(-∞) and ln(∞)

[DrCrowbar]

For instance, say I have

-ln(-∞)​

Does the negative sign on the natural log cancel with the negative sign on the infinity?

Is this true?

-ln(-∞) = ln(∞)​

Thank you

-Drc

 

[tiny-tim]

Hi DrCrowbar!

There is no such thing as the ln of a negative number.

(unless we're allowed complex numbers, in which case eg ln(-1) = πi)

 

[DrCrowbar]

Hi Tim!

Ah... clumsy me. I knew that. I really did. Well, used to...

So if you have ln(-∞), is that essentially ∞ or -∞?

I'm finding the solution for a calculus III improper integrals question. I guess it doesn't matter, though, because either way it diverges (a possible solution).

Thanks again

-Drc

 

[Mark44]

Hi Tim!

Ah... clumsy me. I knew that. I really did. Well, used to...

So if you have ln(-∞), is that essentially ∞ or -∞?

No, it's simply not defined, as tiny-tim said. I'm assuming you're working with real numbers.

Hi DrCrowbar!

There is no such thing as the ln of a negative number.

(unless we're allowed complex numbers, in which case eg ln(-1) = πi)

 

[tiny-tim]

So if you have ln(-∞), is that essentially ∞ or -∞?

No, that's ∞i

 

[DrCrowbar]

Ah, ok.

So ln(-#) is the same as -ln(#)... That makes sense, actually. I had forgotten what the graph of the natural log function looks like.

Thanks guys. It's the first time I've asked a math question online and actually received a correct answer!

-Drc

 

[tiny-tim]

So ln(-#) is the same as -ln(#)

no it isn't!

ln(-#) is an imaginary number (something times i)

if we're only allowed to use real numbers, then ln(-#) doesn't exist!

 

[DrCrowbar]

Oh, ok.

So you can have a negative natural log function (-ln|cscx+cotx| for instance) but you cannot have the natural log of a negative number unless you involve imaginary numbers.

I haven't really seen much of imaginary numbers, but I hear they're used a bit in D.E.

 

[tiny-tim]

yes, they'll be useful later

for now, forget about them

the graph of ln(x) is like the graph of √x …

it simply has no value for x < 0

 

[micromass]

No, that's ∞i

What?? In what context is $ln(-\infty)=\infty i$? What does $infty i$ even mean?

 

[tiny-tim]

oops!

i should have said ln(-∞) = πi + ∞