# Does a negative natural log of a negative number cancel to become a positive log?

1. Jan 29, 2013

### 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

2. Jan 29, 2013

### 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)

3. Jan 29, 2013

### 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

4. Jan 29, 2013

### Staff: Mentor

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

5. Jan 29, 2013

### tiny-tim

No, that's ∞i

6. Jan 29, 2013

### 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

7. Jan 29, 2013

### tiny-tim

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!

8. Jan 29, 2013

### 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.

Last edited: Jan 29, 2013
9. Jan 29, 2013

### tiny-tim

yes, they'll be useful later

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

it simply has no value for x < 0

10. Jan 29, 2013

### micromass

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

11. Jan 29, 2013

### tiny-tim

oops!

i should have said ln(-∞) = πi + ∞