There is always a time when we realize that people telling us that “logarithms of negatives are undefined” is a lie.
With complex analysis, we can derive the following:
As $e^{i\pi} = -1$, $\ln(-1) = i\pi$. Hence, $\ln(-|a|) = \ln|a| + i\pi(2k + 1)$ where $k \in Z$.
This is a pretty general result from complex analysis.
When I first derived this on my own, I curiously typed “$f(x) = \ln(x) + i\pi$” into GeoGebra Graphing calculator, and got this:
$f(x) = \ln(x) + i\pi$" />The first thing I noticed was that every line was at a right angle.
What in the world does this graph mean/imply? And why is GeoGebra plotting a function involving $i$ in the Cartesian plane?
Help is appreciated!
Edit: As Izaak Van Dongen asked, the following are some other functions plotted in GeoGebra:
$a(x) = x + \frac{1+i}{3}$:
$b(x) = x^2 - \frac{i}{2}$:
$c(x)=x^i - 1$:
$d(x) = x^{(2^i)}\sin(ix)i$:
This one is not at right angles.
$f(x) = e^{ix}$:
Here we can see how $e^{ix}$ (with initial condition $f(0) = 1$) can be thought of as walking around the unit circle.
A $\ln$ function can certainly be defined on $\mathbb R$ in a natural way. In fact, one can even define its derivative on $\mathbb R$. For a detailed discussion, see here.