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 $\log$?

  • $\begingroup$ Abstract Duplicate: math.stackexchange.com/questions/90594/… $\endgroup$ – user21436 Feb 12 '12 at 17:07
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    $\begingroup$ 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 "$\ln$" $\endgroup$ – David Mitra Feb 12 '12 at 17:14
  • $\begingroup$ $$ln = log_e$$ $$log = log_{10}$$ (unless the context implies e) $$log = log_2$$ (Algorithms and Computer Science) $\endgroup$ – Inquest Feb 12 '12 at 17:34
  • $\begingroup$ @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? $\endgroup$ – Jim_CS Feb 12 '12 at 18:33
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    $\begingroup$ 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. $\endgroup$ – Chris Janjigian Feb 12 '12 at 18:39

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.



"$\log$" 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.

  • $\begingroup$ Why the down-vote here? If something is objectionable about what I wrote, then telling me what it is would correct the problem. $\endgroup$ – Michael Hardy Feb 13 '12 at 17:57
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    $\begingroup$ I didn't downvote, but I disagree with you position. The issue at hand is the difference between real vs. complex logarithm. Also your claim that base-10 logs are obsolete is ... detached from reality. Have you ever done any signal processing or acoustics? Have you heard of decibels? Ok, I learned to use slide rules before calculators became commonplace, but even so :-). $\endgroup$ – Jyrki Lahtonen Feb 13 '12 at 22:29
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    $\begingroup$ Ironically, you seek support from the conventions adopted by programming language designers, which is more or less the same set of people that designed the calculators. And you denounce the conventions adopted there. $\endgroup$ – Jyrki Lahtonen Feb 13 '12 at 22:32
  • $\begingroup$ It would be exaggerated to say that I "denounced" anything. And I did say "when the topic is mathematics". $\endgroup$ – Michael Hardy Feb 13 '12 at 23:38

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.


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