Log

Things that bother me about the log function.

I was just going to focus on "What base do people mean?" until I realized that there's a lot of ambiguous things about this function.

The first is..."What base do people mean?"  Generally, you have to take it by context of the person you're talking to.  If I'm talking to a computer scientist, it's base 2.  If I'm talking to an engineer, it's base 10.  If I'm talking to a mathematician, it's base e.  We all just say 'log' though.

Next up is, "What's the branch cut?"  Generally, people mean the $-\pi$ branch cut, where log of negative numbers is undefined.  However, over the complex plane, we could mean different possible branch cuts.  Moreover, depending on the surface you're considering the log function over, different things happen, which brings me to...

"What's the domain?"  If it's over $\mathbb{C}$, it makes sense to even ask the previous question.  If it's over $\Re$, then _really_ you're considering the domain $\Re^+$, the positive real axis.  Although, you _could_ define a log function over the negative real axis and leave the positive real axis undefined.  Moreover, there's a particular kind of Riemann surface (in fact, it's _constructed_ so that the following happens) where the log function over _it_ is defined _everywhere_.  Moreover, the question of which domain you're considering is important to answer the question...

"What's the derivative?"  I remember hearing this story of a Physics professor docking points off of a student for drawing the graph of the log function's derivative as being $1/x$, but only on the _positive_ part of the axis.  Technically, the student was 100% correct.  The derivative of $log(x)$ is $1/x$ only on the positive real axis because $log(x)$ itself is only _defined_ on the positive real axis.  Because of this, depending on where the branch cut, domain, and even _base_ you're considering, the derivative will be different!

However, we usually don't clarify these things.  All of this information about the log function is usually easily taken up according to the context of the discussion.