Logarithms

We had our Complex Analysis exam this morning. There were only 5 questions and 2 dealt with a logarithm. So after I spent 20 minutes writing down everything I knew on the exam, I had the rest of the exam to contemplate the logarithm. Now a logarithm is a tricky thing. In junior high you learn about it and it seems strange but innocuous, like a polky-dotted duck or something. It acts like a function and quacks like a function so who cares if it's a little wierd, right? Well, as it turns out, the logarithm is hiding some serious identity crises. In the complex plane it isn't just a function, it's simultaneously a whole bunch of functions anywhere but none of them everywhere. What's worse, this isn't some random identity crises that we can just ignore by relegating the logarithm so red-headed stepchild status among functions. A lot of important functions that look perfectly nice from far away have logarithm-type issues. Like all these functions are actually crazy but only the logarithm is honest about it.