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I'm confused how do i can justifying to my students the difference between the two Notation “$\log x$” and “$\ln x$” however i checked the definition here in wikipedia. Really many people understand that the notation “$\log x$” must be the decimal logarithm (base $10$). But what I think is $\log x$ can be written as $\log_{e}x$ which is $\ln x $. The difference occurs only when I put the base $a$ as $\log_{a}x$ different from $e$.

My question here is:

Should the notation “$\log x$” mean the decimal logarithm and how can I give the correct notation with the correct definition to my students?

Thank you for any help

egreg
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zeraoulia rafik
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    yeah, it used to be $lg$, but the mathematical society was sued by a certain electronics manufacturer. – Asinomás Dec 25 '16 at 22:05
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    It depends what part of mathematics you are working in and at what level. Most of the time "$\log x$" will mean base $e$ in higher mathematics whereas at lower levels the notation $\ln$ is often used. – Edward Evans Dec 25 '16 at 22:06
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    @Jorge: In some areas of mathematics $\lg x$ is commonly used for $\log_2x$. – Brian M. Scott Dec 25 '16 at 22:08
  • I think if i said log x it must be to inform my student that i meant for example base 2 or 10 or e – zeraoulia rafik Dec 25 '16 at 22:10
  • @Jahambo99 What do you mean by "higher" and "lower" parts of mathematics ? – Jean Marie Dec 25 '16 at 22:10
  • @JeanMarie I guess I mean the distinction between university and school mathematics. – Edward Evans Dec 25 '16 at 22:12
  • @Brian M. Scott The use of lg x is now very very scare. You don't find it any more in publications for example. – Jean Marie Dec 25 '16 at 22:12
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    @Jahambo99 In my (European) country, almost everybody in Mathematics uses now ln(x) either in secondary school or University. – Jean Marie Dec 25 '16 at 22:14
  • @JeanMarie Strange! I was told in my british university that $\log$ is much more widely used for base $e$ in mathematics post-school. Maybe it's just the age of the lecturers. – Edward Evans Dec 25 '16 at 22:17
  • @Sebastiano Hiiiiiiiii and Merry Christmas to everybod. When I was a boy I read about old books $\Log(x)=\log_{10}(x)$. –  Dec 25 '16 at 22:21
  • @JeanMarie: I wouldn’t know about publications, but I certainly encounter it often enough in online course notes and arXiv papers. – Brian M. Scott Dec 25 '16 at 22:22
  • @Sebastiano: I think Log denoted ln — supported by the fact that it's Log which is used in Complex Analysis. I've always seen log for the decimal logarithm. – Bernard Dec 25 '16 at 22:28
  • @Bernard You can see at the link http://www.sapere.it/enciclopedia/logaritmo.html. Logaritmo decimale. Sorry but it is in Italian. –  Dec 25 '16 at 22:33
  • @Bernard There are many link for $\Log$: http://www.giocomania.org/pagine/29239/pagina.asp. –  Dec 25 '16 at 22:38
  • Related (dup?) http://math.stackexchange.com/questions/1221039/which-is-more-preferable-to-write-logx-or-lnx http://math.stackexchange.com/questions/424484/what-does-ln-x-or-log-x-mean/424488 http://math.stackexchange.com/questions/369049/meaning-of-log – leonbloy Dec 25 '16 at 22:39
  • In high energy physics we also frequently use $\log$ to mean the natural log. I once had a math processor state that this practice is more coon in higher maths. It had never bothered me, and have inadvertently used it while tutoring students due to forgetting they might use different notation. – Triatticus Dec 25 '16 at 23:14
  • Related as well: (http://hsm.stackexchange.com/questions/2762/why-is-the-natural-logarithm-represented-by-ln) – Jean Marie Dec 26 '16 at 00:12

3 Answers3

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It honestly doesn't matter as long as you are clear. For example, wolfram uses $\log(x)$ even when it means the natural log but clarifies this in a footnote. I personally prefer $\ln(x)$ (when teaching highscool) because it requires no further clarification.

At the end of the day it doesn't really matter; definitions and notation are two separate things. Just pick one and be consistent. Don't penalize your students for choosing certain accepted notations over others as long as they are clear about their intent i.e. $\log_e(x)$ is as valid as $\ln(x)$ which is as valid as $\log(x)$ where $\log(x)$ is the natural logarithm.

Being too picky about these things can cause students to associate Mathematics with some suffocating and obscure set of rules and rituals that really have nothing to do with Mathematical thinking.

gowrath
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    @gowrath, really all students in my country belive that log x must be meant logarithm decimal without any precdent definition or indication , that is the problem !!! – zeraoulia rafik Dec 25 '16 at 22:14
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    @user51189 There are very few cases I can think of where it wouldn't be clear from the context which one they mean. I teach a lot of international students so I see huge varieties in notation. If the majority of your students have a regional preference, then let them stick with the convention that $\log(x)$ is the base 10 logarithm and $\log_e(x)$ or $\ln(x)$ is the natural log. It is true in early years that the base 10 logarithm seems more important and so $\log(x)$ is preferred by students. It really depends what you think is best for your students in terms of learning and understanding. – gowrath Dec 25 '16 at 22:31
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In pure mathematics there is one and only one logarithm $\log$ defined to be the inverse function of $\exp$. Personally I do not like the notation $\ln$ since for example in real, complex or functional analysis you will never encounter any other logarithm than the natural one. The most clear notation I've encountered in the Book Analysis I by Vladimir A. Zorich: He uses $\log$ to denote the natural one and $\log_a$ for $a>0$ is the inverse function of the exponential mapping $a^x := \exp(x\log a)$. See, even here we use just the most fundamental transcendental functions.

But this is my way of how I would define it. There are several possibilities and most of the time definitions are a matter of taste. The most important part is to be consistent.

TheGeekGreek
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  • Thanks and this is what i think , i hate using ln x and I use the notation log to mean base e with log_a(x) for a different from 0 to define any base – zeraoulia rafik Dec 25 '16 at 22:22
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    Maybe it would be helpfull if you add what "my students" are, i.e. there are other almost standard conventions in computer science. – TheGeekGreek Dec 25 '16 at 22:26
  • students high school level , the inconvinient which i found when students use the notation "ln" is : calculation of limit for example , as student mixed between lim and ln , lim ln(x) it's not gud as lim log(x) , the latter let students forget ln because they are seen it as limit " – zeraoulia rafik Dec 25 '16 at 22:28
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Calculators denote the the logarithm in base 10 as "Log" and the natural logarithm by "Ln". Teachers then need to refer to the logarithms in the way these are indicated by the calculators as students are first exposed to simple pre-calculus problems involving exponentials and logarithms where they need to use their calculators, like powers of two and then trying to find for which power of two the outcome is some given number etc. etc.

When students are exposed to calculus, they will have gotten used to "Ln" for natural logarithm, so the teachers will not switch to the official notation. Only at University will students have to get used to "log" for the natural logarithm because that's the way it's used in the scientific literature.

Count Iblis
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  • Infact. When I had 17 years, "Log" was $\log_{10}(x)$. –  Dec 25 '16 at 22:40
  • I think that is mostly the problem as students nowadays not really understand what they are doing. I mean, I see it quite often that a student just types the question one to one in a calculator, i.e. solving an equation, but does not even understand what the purpose is. – TheGeekGreek Dec 25 '16 at 22:44