Why are $\log$ and $\ln$ being used interchangeably? A definition for complex logarithm that I am looking at in a book is as follows -
$\log z = \ln r + i(\theta + 2n\pi)$
Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the log will make a difference to the answer.
It also says a few lines later $e^{\log z} = z$.
Yet again I don't see how this makes sense. Why isn't $\ln$ used instead of $\log$?
 A: Often in math books the base of $\log$ is just assumed to be $e$. 
In this case it looks like the reason they are using $\log z$ instead of $\ln$ is to differentiate between when it is a complex function versus when it is a real function.
http://en.wikipedia.org/wiki/Complex_logarithm
A: "$\log$" with no base generally means base the base is $e$, when the topic is mathematics, just as "$\exp$" with no base means the base is $e$.  In computer programming languages, "$\log$" also generally means base-$e$ log.
On calculators, "$\log$" with no base means the base is $10$ because calculators are designed by engineers.  Ironically, the reasons for frequently using base-$10$ logarithms were made obsolete by calculators, which became prevalent in the early 1970s.
A: This is one of those cases where two different notations developed in slightly different applications, and both are commonly used enough that one has not been adopted over the other. In some fields of engineering, $\log$ means $\log_{10}$, in math it usually means $\ln$, and in computer science it often means $\log_2$ (when it matters). Another example of this kind of notational difference is found in boolean algebra. While a mathematician might write an expression as $(a\land b)\lor(\lnot c\land d)$, an electrical engineer would probably write this as $ab+\overline{c}d$. Take also the Leibniz versus Newton calculus notations. It is what it is, I find that the best practice is just to know various different notations and guage what is most appropriate in context.
