Wednesday, March 6, 2019

logarithms - 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:


enter image description here


However, Wolfram|Alpha is graphing it like this:


enter image description here


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 $\mathbb{R}$?


Answer



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


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