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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 $\log$?

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    Abstract Duplicate: http://math.stackexchange.com/questions/90594/the-difference-between-log-and-ln2012-02-12
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    As far as I can surmise, when speaking of the logarithm of a complex variable (inverse of the complex exponential), you write "$\log$", and when speaking of the (natural) logarithm of a real variable, you write "$\ln$"2012-02-12
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    $$ln = log_e$$ $$log = log_{10}$$ (unless the context implies e) $$log = log_2$$ (Algorithms and Computer Science)2012-02-12
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    @David Mitra - Normally log on its own means "$\log_{10}$" so when speaking of the logarithm of a complex variable you write "log" and it is implied that you are actually using "$\log_{e}$"...is that correct?2012-02-12
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    This is fairly standard notation in complex analysis texts. The reason is that you have to define a branch cut of the logarithm, which needs to be distinguished from the real valued logarithm used implicitly in the construction of the branch cut. The issue here is not that the base of the logarithm has changed, but that it is important to specify which branch of the complex logarithm you are talking about and to differentiate that object from the real natural logarithm which is unambiguously defined.2012-02-12
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    @Nunoxic : Please: writing \ln and \log in TeX with a backslash not only prevents italicization but also causes proper spacing conventions to be respected. Those are the correct spellings. With "log x", you get this: $log x$. With "\log x", you get this: $\log x$.2012-02-12
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    In mathematical contexts at or above calculus, $\log$ with no base written is usually the natural logarithm, as that's the only log that has special properties (the "common" logarithm, or base-10 logarithm, doesn't have any particularly special mathematical properties, just convenience relative to our writing numbers in base 10).2012-02-12
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    Conventions differ. Most mathematicians rarely use any base for logarithms other than $e$. I was surprised to find that there is supposedly an international standard: < http://en.wikipedia.org/wiki/ISO_31-11#Exponential_and_logarithmic_functions >. Does anybody follow this?2012-02-12
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    @Numoxic: You state the conventions used in a particular book. You seem to imply that these same conventions are widely used in mathematics.2012-02-12
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    StackExchange readers may be interested in an experience I had with a high school teacher on this matter. See the bottom part of the following post: http://mathforum.org/kb/message.jspa?messageID=7188533 I later posted a follow-up that included more about this teacher (who I otherwise think highly of): http://mathforum.org/kb/message.jspa?messageID=68736212012-02-13

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