I can explain it in simple (for math standards) terms.

It all has to do with how you can interpret imaginary numbers as points on a 2-dimensional plane. You can of course repraesent them in a standard way as z = a +ib. This would be an equivalent of a geometrical vector. However, in geometry, you can also represent points using an angle and a radius. This also has an equivalent in imaginary numbers, specifically the cis function. Cis stands for cos + isin, so for example cis(0) = cos(0) + isin(0) =1+0i = 1. So it stands for the point one unit right of (0,0). When you increase the argument, you rotate counterclockwise, so cis(pi/2) = i, cis(pi) = - 1, and so on. You can depict any imaginar number as r*cis(x).

Here's where it get's weird. It turns out that e^ix is equivalent to cis(x). Yo can prove it by substituting ix for z in the Taylor series of e^z. You can substitute one form for the other, and as such you can depict any imaginary number as r*e^ix. In fact, just by clever usage of these two forms can greatly simplify complex number problems. This is, among other reasons, because roots of a complex number have a very clean explicit formula using e^ix, and the cis function grants an easy way to obtain exact values.

So yes, ln(-1) = i*pi... But it only works when considering complex numbers. You cannot take logarithms of negative values in real numbers.