Graph of $y=x^x$ for $x<0$

Question

I have been wondering about the graph of $y=x^x$. Most graphing calculators will quite happily graph it up to $0$, but after that they don't do anything else. Basic calculation suggests that, while for some points ($x=-\frac 12$) there are no real number solutions, for others ($x=-1,-2,-3$ etc.) there are solutions. Why, if at all, do the graphing calculators stop at $0$, and can anyone produce a graph of the real number solutions of $y=x^x$ past the $0$ point? Thanks.

Answer 1

Note that $f(x)=x^x = e^{x\ln(x)}.$ Since $e^x>0$ for $x\in\mathbb{R}$, there exists no real logarithm of negative real numbers.

However, by the Euler identity $e^{\pi i}=-1$, therefore you could say that "$\ln(-1)=\pi i$", which is a complex number. The problem with that is that the exponential function is periodic, i.e. $e^{(2k+1)\pi i}=-1$ for every odd number $2k+1$, $k\in\mathbb{Z}$. Therefore you could just as well say that "$\ln(-1)=3\pi i, -\pi i, \dots$". See multivalued function, complex logarithm.

The graph which you see in Wolfram Alpha gives you for negative real values the real and imaginary part corresponding to using the principal branch (one choice of values for the logarithm which is somehow canonical) of the logarithm.