Logs of negative numbers.

Fallen Seraph

My maths book, she tells me you can't get the log of a negative number! Yet-2^3=-8
=> log(-2)-8=3....

That makes perfect sense to me... So what am I thinking wrong?

Crumbs

Fallen Seraph wrote:

My maths book, she tells me you can't get the log of a negative number!

Unless your base is negative, as in your example.
To the base 10 or base e, you can't.

Edit: Actually, thinking about it more, it's a bit more complicated than that. There are reasons you can't use negative bases due to it leading into complex numbers.

JackieChan

Edit: Actually, thinking about it more, it's a bit more complicated than that. There are reasons you can't use negative bases due to it leading into complex numbers.

This would only cause a problem if you were working in non-complex dimensions(eg Reals,Integers,Rationals).

incisor71

Crumbs wrote:

Unless your base is negative, as in your example.
To the base 10 or base e, you can't.

Edit: Actually, thinking about it more, it's a bit more complicated than that. There are reasons you can't use negative bases due to it leading into complex numbers.

I know this isn't directly related to the original logarithms question, but because exponents and logarithms are complementary operations, I thought I'd drop this into the discussion...

I stumbled across a footnote in one of the O'Reilly books "Learning Perl" (strangely enough) that, if you want to avoid the complex numbers domain, you may raise a negative value only to an integer-value exponent, such as the example cited above (-2)^3 = 8 or (-3.5)^3 = -12.25.

If you wanted to raise a negative value to a non-integer exponent like so: (-2)^2.5 you'd have to split up the operation as follows:

(-2)^2.5 = (-2)^2 * (-2)^0.5
= 4 * (-1)^0.5 * (2)^0.5

and since (-1)^0.5 = i, then the result is

= 4 * (2)^0.5 i
......which is a purely imaginary number.

I don't know how one would handle computation of (-1)^0.3 or some other exponent that's not a multiple of 0.5, but no doubt some enlightened soul can inform me/us.

So it appears that one can obtain the logarithm of certain negative numbers, but even then only under very restricted conditions.

silverside

you can get a log of anything

say x = ||x|| (cos theta + i sin theta)

remember your argand diagrams

then log x = log ||x|| + i (theta + 2n pi ).

http://mathworld.wolfram.com/Logarithm.html

of course to raise things to powers, do the same in reverse. the basic idea (for complex numbers and integer powers) is in the leaving cert course , from what i remember

incisor71

silverside wrote:

you can get a log of anything

say x = ||x|| (cos theta + i sin theta)

Ah yes, Euler's identity... forgot completely about that one (how embarrassing, seeing as my first Masters degree involved a lot of DSP)! Thank you.

incisor71

silverside wrote:

of course to raise things to powers, do the same in reverse. the basic idea (for complex numbers and integer powers) is in the leaving cert course , from what i remember

Euler stated that cos theta + i sin theta = exp (i theta).

Tell me if this looks to you (re-using the previous example, where I want to raise -2 to the power of 2.5):

Find the mag/arg, or polar representation, of -2, which in this case is:

2*exp (i*PI).

or -2, at an angle of PI on the Argand diagram.

Raise both mag and arg to the power of 2.5, which gives:

mag = 2^2.5 = 4*[2^0.5]
arg = {exp (i*PI)}^2.5 = exp (i*2.5PI) = exp (i*0.5PI) = cos (0.5PI) + i sin (0.5PI) = = 0 + i

Combine them to yield:

mag/arg = {4*[2^0.5]} i

= 5.6568 i as before. Seems ok to me!