I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this:
However, Wolfram|Alpha is graphing it like this:
As you can see, there is a "real" range in the region $(-\infty, 0)$, and an imaginary part indicated by the orange line. Is there a part about log graphs that I am missing which would explain why Wolfram|Alpha shows the range of the log function as $\mathbb{R}$?
$\ln(x)$ is formally defined as the solution to the equation $e^y=x$.
If $x$ is positive, this equation has an unique real solution, anyhow if $x$ is negative this doesn't have a real solution. But it has complex roots.
Indeed, $\ln(x)= a+ib$ is equivalent to
$$x= e^{a+ib}= e^{a} (\cos(b)+i \sin (b)) \,.$$
If $x <0$ we need $e^{a}=|x|$, $\cos(b)=-1$ and $\sin(b)=0$.
Thus, $a= \ln(|x|)$ and $b=\frac{3\pi}{2}+2k\pi$....
As well as being an $\mathbb{R^+} \to \mathbb{R}$ function, the logarithm can also be extended to a multi-valued complex function. Wolfram Alpha interprets the logarithm as the complex logarithm, then restricts it to real line again for graphing. See http://enwp.org/wiki/Complex_logarithm for a full graph of the complex logarithm.
