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Perhaps am the only one wondering that...

Why the notation of the natural logarithm changes according to the reference is used. So, is the following TRUE? $$\ln{\left| x \right|} = \ln{x} = \log{x}$$

I always thought that $\log{x}$ was the notation convention to write the logarithm function with base $10$. So why can we write the previous equation!?

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  • $\begingroup$ In mathematics past elementary calculus (and often even there), base $10$ logarithms are hardly ever used. Also, in many applications the base doesn't matter, such as when quotients of logarithms are involved or when using logarithms to describe growth rates (where a multiplicative constant doesn't change things). $\endgroup$ Oct 26, 2015 at 20:59
  • $\begingroup$ @DaveL.Renfro But why use the notation $\log$ rather than $\ln$. It isn't because it's more faster to write!? $\endgroup$
    – hlapointe
    Oct 26, 2015 at 21:06
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    $\begingroup$ I think it's because the widespread use of the $\ln$ notation is relatively recent (50 or 60 years?) and mostly restricted to "school mathematics", which forms only a few years in the life of a practicing mathematics user. Personally, I always use $\ln$ when I intend natural logarithm because there's no ambiguity, but if you look through advanced undergraduate and graduate level complex variables textbooks, you'll find $\log$ used almost universally. $\endgroup$ Oct 26, 2015 at 21:15
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    $\begingroup$ Related to this topic, you might be interested in the following 20 October 2009 math-teach post archived at Math Forum. $\endgroup$ Oct 26, 2015 at 21:30
  • $\begingroup$ @DaveL.Renfro, can you convert your comment to answer? To push this question off unanswered queue! $\endgroup$ Nov 14, 2015 at 15:58

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(comment from 3 weeks ago, with an additional paragraph, given as an answer, as requested by @Jesse P Francis. I'll add more later if I find that I have anything more to say that might be of interest.)

I think it's because the widespread use of the $\ln$ notation is relatively recent (50 or 60 years?) and mostly restricted to "school mathematics", which forms only a few years in the life of a practicing mathematics user. Personally, I always use $\ln$ when I intend natural logarithm because then there will be no possible ambiguity, but if you look through advanced undergraduate and graduate level complex variables textbooks, you'll find $\log$ used almost universally for natural logarithms.

In most papers and books before the 1880s (roughly speaking), logarithms were indicated by a lowercase "L" letter (and maybe sometimes an uppercase "L"), which I've personally found sometimes difficult to parse when digits are involved, especially the digit $1.$ For example, if you spend much time looking at older literature on infinite series (as I found myself doing several months ago for a project that I expect to eventually post in the History of Science and Math Stackexchange for an answer to an old question), you will see this "lowercase L notation" all over the place.

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