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In complex analysis I came across three types of logarithms namely $\ln$, $\log$ and $\text{Log}$. What is the difference between the three?

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    $\begingroup$ Doesn't the text where you found those notations explain their meaning? $\endgroup$ – hmakholm left over Monica Mar 31 '16 at 20:02
  • $\begingroup$ $\log_ba$ is the logarithm of $a$, base $b$. When no $b$ is specified, $b=e$, This is the natural logarithm. Another way of denoting the natural logarithm is $\ln n$ $\endgroup$ – Mastrem Mar 31 '16 at 20:03
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    $\begingroup$ Typically a graduate math text on complex analysis will use $\log$ to refer to the natural logarithm. In computer/numerical analysis contexts it may be more common to use $\ln$ for the natural logarithm and to use a subscript on $\log$ or $\operatorname{Log}$ to distinguish logarithms to other bases (such as two or ten). Your text should indicate the definitions if multiple notations are used. $\endgroup$ – hardmath Mar 31 '16 at 20:05
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    $\begingroup$ I would think $\ln$ and $\log$ are the same and that $\operatorname{Log}$ refers to the principal logarithm as listed here en.wikipedia.org/wiki/Complex_logarithm under "Definition of Principal Value". $\endgroup$ – Joe Johnson 126 Mar 31 '16 at 20:05
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    $\begingroup$ It is a perfectly reasonable question. $\endgroup$ – copper.hat Mar 31 '16 at 20:09
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I thing if $z=re^{i\theta}$ then $\log z=\ln r+i\theta$. Now if in the arguement $\theta ,n=1$ then "$\log$" becomes "Log". Also "$\ln$" is generally used with real part of complex number as per the notations used by C.V.Chirchill

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    $\begingroup$ What does "in the arguement $\theta, n = 1$" mean? $\endgroup$ – ruakh Mar 31 '16 at 21:53
  • $\begingroup$ There may be a language problem in narrating the answer but that doesnt mean the answer is wrong.Actually the argument $\theta$ satisfies $logre^{i\theta}=lnr+i\theta$ so does $2n\pi+\theta$ $\endgroup$ – Rayees Ahmad Apr 1 '16 at 0:34
  • $\begingroup$ Yeah, I don't know why people downvoted your answer, it is kind of unfair. $\endgroup$ – KKZiomek Apr 1 '16 at 1:51
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    $\begingroup$ There seems to be no reference as to what this "$n$" thing is in the answer. It's apparently a part of the argument, so…what does that mean? Your comment makes up for what lacks in the answer though, namely, you explain why we should bother about any $n$. Honestly, upon first reading the answer, I thought you meant that the $n$ in "$ln$" should be 1, which is clearly wrong. Nowhere in either the question or in your answer is it explained that an $n$ variable even exists, so I assume that is where the downvote came from. "language problem", not so much, rather an "explain yourself" one… $\endgroup$ – A.Sh Apr 1 '16 at 10:33
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    $\begingroup$ I have a BS in Mathenatics and I do not understand this answer at all. Am I the only one? I think it would help a lot if proper markup were used for the mathematics in this answer. $\endgroup$ – Todd Wilcox Apr 1 '16 at 11:05
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  • ln is always a natural logarithm (of base $e$).

  • log if it has a base like $\log_5x$ then the base is the specified one, otherwise it is either base $e$ or base $10$. It depends on people. Some people (mostly high school people) use logarithms without a base as base $10$ and other people as base $e$. It is most often used as a natural logarithm though but you have to watch out. If you see a ln button on a calculator, then log without base is guaranteed to be base 10 on the same calculator.

(Edit: As @ClementC. pointed out, in computer science, $log$ without a base most often means a binary logarithm.)

  • Log stands for complex logarithm in its principal branch. The principal branch has its imaginary part in the interval $(−π,π]$. (Basically it's the inverse of the complex exponential function with its imaginary part in that interval.)
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    $\begingroup$ In mathematics $\log$ tends to always mean the natural log. $\endgroup$ – Edward Evans Mar 31 '16 at 20:07
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    $\begingroup$ $\text{Log}$ is often defined as the principal branch of the complex logarithm. $\endgroup$ – Winther Mar 31 '16 at 20:09
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    $\begingroup$ In computer science, more often than not $\log$ is used for base $2$. (Sometimes, it is specified in the "preliminaries" section of the paper.) $\endgroup$ – Clement C. Mar 31 '16 at 20:12
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    $\begingroup$ Also, until around the 1970s, log usually designed the decimal log, simply because it was ubiquitous in numerical computations "by hand", with a table of logarithms or a slide rule. This use is really what made the decimal log popular in the first place, I guess. $\endgroup$ – StayHomeSaveLives Mar 31 '16 at 20:29
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    $\begingroup$ Also, in computer science $\log$ often means just "any logarithm" - the base doesn't matter as logarithms are all equal up to a constant factor. $\endgroup$ – JohannesD Apr 1 '16 at 8:33
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These are notations, sometimes ambiguous, to denote potentially different types of logarithms, which depend on the language of origin (Russian, German, French), see for instance where $\lg$ is used, including number theory, since sometimes $\log_2 x $ denotes the iterated logarithm: $\log{\log{ x }}$.

Notation $\ln x$ (almost) unambiguously denotes the natural logarithm $\log_e x$ (latin: logarithmus naturalis), or logarithm in base $e$. In French, I used to believe the "n" stood for "népérien", from Neper or Napier.

The notation $\log x$ should be the adopted notation for the natural logarithm, and it is so in mathematics. However, it often represents the "most natural" depending on the field: I learned it as the base-$10$ logarithm ($\log_{10} x$) at school, and it is often used this way in engineering (for instance in the definition of decibels): enter image description here And it may also represent a base-$2$ logarithm (binary logarithm) in binary calculus. The latter is sometimes denoted lg, ld (logarithmus dualis), or lb.

$\operatorname{Log} x$ normally is the principal value for a complex number, with imaginary part in the interval $(-\pi,\pi]$. But, while I learned at school $\log x$ for $\log_{10} x$, $\operatorname{Log} x$ was used (wrongly) to denote the natural logarithm. Such notations can be found for instance in Calcul differentiel et integral. Tome 1, 1998, N. Piskounov, page 58 sq.

The History of logarithms wikipedia page is worth reading, if only to learn the origin of the name (a number indicating a ratio: logos, proportion, and arithmos, number), and John Napier's book: Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).

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