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 $\log$?

"$\log$" 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.
