Ln(-1) = 0 ?

1. Mar 6, 2007

alba_ei

ln(-1) = 0 ???!

supposed that we have

$$\ln(-1)$$

then
$$\frac{2}{2}\ln(-1)$$

so

$$\frac{1}{2}\ln(-1)^2$$

this is equal to

$$\frac{1}{2}\ln(1)$$

and if this is equal to 0 the we can say that

$$ln(-1) = 0$$

is this right , wrong, are there any explanations for this?

2. Mar 6, 2007

cristo

Staff Emeritus
The rule of logarithms is a lnx=lnxa. In this case, a=1-- you cannot split it into a fraction and then only take the numerator!

If you look at the logarithm graph, you will see that the function is not defined for negative x.

3. Mar 6, 2007

morphism

It's wrong. ln(-1) is no longer a real number, so you can't treat it like one. This is like saying sqrt(-1) = (-1)1/2 = (-1)2/4 = ((-1)2)1/4 = 11/4 = 1.

4. Mar 6, 2007

morphism

Actually, that step is perfectly valid in general - (a/a) ln(x) = 1/a ln(x^a), i.e. when everything is defined. Your next line explains why it's not valid here:

5. Mar 6, 2007

cristo

Staff Emeritus
Course it is; sorry!

6. Mar 6, 2007

JustinLevy

Really? Why can't one say that ln(-1)= i pi, for e^(i pi) = -1.
Or is there something wrong with that line of logic?

7. Mar 6, 2007

Semo727

logarithm is defined also for complex numbers.
ln(z)=ln(abs(z))+i*arg(z), where z is complex number, abs(z) is complex norm of complex number z, and arg(z) is its argument.
So if -1 is treated as complex number -1+0*i, expression ln(-1) gives sense, but the identity a*ln(z)=ln(z^a) is no longer true.

8. Mar 6, 2007

cristo

Staff Emeritus
To satisfy the pedants, I shall re-phrase my above answer. The natural logarithm function, whose argument is a real number and to whom we can apply the standard laws of logarithms, is not defined for negative real numbers.

9. Mar 6, 2007

JonF

for complex z: Ln(z) = ln(|z|) + i*Arg(z)

so ln(-1) = ln(|-1|) + i*Arg(-1) = i*pi

Last edited: Mar 6, 2007
10. Mar 6, 2007

morphism

Really? If you were attempting to define the principal branch of Ln, then it ought to be Ln(z) = ln(|z|) + iArg(z), where ln is just the natural logarithm on the reals.

In this case, we have Ln(-1) = ln(|-1|) + iArg(z) = i*pi.

11. Mar 6, 2007

JonF

eh forgot the ln, fixed

12. Mar 7, 2007

HallsofIvy

Staff Emeritus
Cristo said, "for negative x". Since the complex numbers are not an ordered field, there are no "negative" complex numbers. Cristo was clearly talking about real numbers.

13. Mar 8, 2007

Gib Z

$$\log_e -1 = i\pi + 2ki\pi, k\in \mathbb{Z}$$ Case Closed.

14. Mar 8, 2007

murshid_islam

is the logarithm of complex numbers defined for any other base except 'e'?

15. Mar 9, 2007

AlphaNumeric

Yes, you can still change between various different bases for your logarithms in the same manner as you do for Real numbers.