Graph of a Log Function

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 $\mathbb{R}$?

• The graph on the negative side is the graph of the log as a complex valued function. Note that "$\log(-x)=\log (x) + \log(-1)$", and $\log(-1)$ is a complex number. You can figure easily which one if you are familiar with the Euler formula $e^{i \pi} =-1$. – N. S. Jan 29 '12 at 23:54

$\ln(x)$ 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 $b=\frac{3\pi}{2}+2k\pi$....

• Excellent! Thank you for the detailed explanation! – Oliver Spryn Jan 30 '12 at 0:07

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.

• FYI, the link did not work for me. – NoChance Jan 30 '12 at 0:29