# Talk:E (mathematical constant)


### E (mathematical constant)
The redirect E (mathematical constant has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 9 § E (mathematical constant until a consensus is reached.  Utopes (talk / cont) 01:48, 9 April 2024 (UTC)

We had a discussion as to whether we should say, "e is the base of natural logarithms" or "e is the base of the natural logarithm" and no clear consensus was reached. Nevertheless I thought the agreement was to use the normal English phrase, "base of natural logs." Now this has been reopened by an editor who not only used the phrase, "base of the natural logarithm" but reverted the correction without an explanation. Can we settle this please?
There are an infinity of natural logarithms but they all have one thing in common. If they are natural logarithms then their base is e. If you say 'the natural logarithm' then you mean just one value. But which value? For example the natural logarithm of 2 is 0.69314718…. Its base is e but why choose ln(2) rather than ln(3)? It is much easier to indicate that the base of all natural logarithms is e. Of course the base of the natural logarithm function is e but why use stilted English instead of the common phrase. OrewaTel (talk) 09:59, 15 April 2024 (UTC)

There are an infinity of natural logarithms (one for each real number), and "the natural logarithms" means all of them considered together. So "e is the base of the natural logarithms" is a mathematically correct formulation, but in "e is the base of the natural logarithm", the singular is definitively wrong. If considered formally, the sentence "e is the base of natural logarithms" means "the base of some natural logarithms". So it is mathematically less accurate, although often used coloqually. As I am not a native English speaker, I am not able to decide whether this colloqual usage is dominant or not. D.Lazard (talk) 10:35, 15 April 2024 (UTC)
You might want to note, as you did in the archive, that "logarithm" also refers to a unary operation, not just a single number, at least in its common use in English. Tito Omburo (talk) 10:40, 15 April 2024 (UTC)
In fact, the archive had pretty solid consensus for "base of the natural logarithm", but I would add that this is very standard in introductions on the subject.  (For example, Stewart's Calculus.)  The article itself uses the phrase "the natural logarithm" numerous times.  I would argue also that "base of natural logarithms" is incorrect or misleading, because there is only one natural logarithm, namely ln.  Tito Omburo (talk) 10:37, 15 April 2024 (UTC)
The phrase "the natural logarithm" is used 6 times in the article. The first time is the one we are discussing. The second time it is clearly talking about the natural logarithm function. ("The natural logarithm is the inverse of the exponential function.") Third and fourth times are specific instances. (Which is correct as we can talk about the natural logarithm of a particular number.) The fifth and sixth time explicitly talk about the natural logarithm function. So I ask again, for which specific number are we taking the natural logarithm?
In Stewart's Calculus the chapter heading on page 64 is "Natural Logarithms". The following clause is in the introduction, "The logarithm with base e is called the natural logarithm". Thereafter whenever Stewart talks in general he uses 'natural logarithms' but when talking about a logarithm of a specific number he uses 'the natural logarithm'. Sometimes he writes 'the natural logarithm function'.  OrewaTel (talk) 11:24, 15 April 2024 (UTC)
In common parlance, "the natural logarithm" refers to the function.  This is not controversial.  Remarks like "for which specific number are we taking the natural logarithm" suggests that you fail to understand this.  I suggest that you make an attempt.  You can compare ghits if you like [1] versus [2], which show that the former is far more common.  Tito Omburo (talk) 11:51, 15 April 2024 (UTC)
In common parlance, one talk of "the natural logarithm of 3" not of "the value at 3 of the logarithm". For making correct (but rather pedantic) the last formulation, one must add "function" at the end. Also, for centuries, everybody used "table of logarithms", not "table of the logarithm". "Logarithm" is often used as an abbreviation for "logarithm function", but this is an abbreviation that should be used only when it is clear from the context that one talks of the function.
Nevertheless, the counts of ghits are not  WP:reliable sources. Similar counts at Scholar Google are more significant, and provide a slight advantage on plural vs. singular.
In any case, you changed (without any discussion) on the 18 March a previous formulation that was previously discussed. I have thus restored the previous formulation.
By doing this, I remarked that on the 23 March, you removed from the lead all definitions of the constant. This may be confusing for many readers, as they should be able to recognize immediatly the common definitions that thy may have already encountered. So, I have restored the most basic definitions in the lead, leaving in § Definitions the definitions that are more technical or require external concepts, such as integrals, inverse functions, etc. I cleaned up also § Definitions for giving the definitions at the beginning of each paragraph, and removing explanations of auxiliary concepts that are linked to. D.Lazard (talk) 14:53, 15 April 2024 (UTC)
 I don't really object to the new lede, less the lengthy second paragraph from the old version.  I still feel that "base of the natural logarithm" is more than adequately supported, both by regular usage in mathematics and by sources.  And scholar actually shows a slight preference for the singular rather than plural: [3], [4].  This, together with the massively more prevalent ghits of the singular, strongly suggests that the singular is typical in English usage.  (See also, in complex variables, one talks of the "branch of the logarithm", also, one talks of the "sine and cosine", the "natural exponential", and so forth, it being understood that one is referring to functions.)  Note that at one point, you had agreed that "natural logarithm" refers to the unary operator, rather than just its values.  As had User:Trovatore, User:Ancheta Wis, and User:Quantling.  So, I'm hardly out on a limb: the earlier consensus clearly favored the singular.  As Trovatore wrote:
Natural logarithms aren't objects that have a common "base". Natural logarithms, as objects, are just numbers. On the other hand the natural logarithm, in context, is a type of logarithm, and different types of logarithms have different bases. So I disagree with your claim about the "correct general sentence"; the best short solution is in fact "base of the natural logarithm", with no plural.
The direct answer to "which specific logarithm" is "the 'natural' one", as opposed to, say, the "common" one or the "binary" one. Here the "ones" are not numbers, but rather logarithms, a logarithm in this context being the logarithm function to a particular base, rather than a value of that function.
It's difficult to disagree with this.  Do you agree? If so, I will restore the old consensus singular.  Tito Omburo (talk) 15:21, 15 April 2024 (UTC)

I've put back in the "previous version that was discussed", per your post, since the outcome of the discussion was "the base of the natural logarithm", with the singular, which was also the earlier consensus version prior to this edit you had made, contrary to any clear consensus I was able to find, with one other editor User:Sapphorain making good arguments for the status quo. Tito Omburo (talk) 15:32, 15 April 2024 (UTC)
Note also the singular in the French article fr:Logarithme népérien. Tito Omburo (talk) 15:48, 15 April 2024 (UTC)
"Base of the natural logarithms" sounds better to me, but it turns out that both phrases are used these days in reputable sources,[5][6] and the singular seems to be more popular now. So either one is fine. --Macrakis (talk) 14:53, 15 April 2024 (UTC)
The French article on e starts, "Le nombre e est la base des logarithmes naturels." There is an article "Logarithme népérien" but that makes it very clear that it is talking about the 'fonction'. So to obviate a silly edit war, I'm adding the word function to the lead. It is stilted English but at least it is correct and makes sense. OrewaTel (talk) 02:51, 17 April 2024 (UTC)
The lede of that article also includes the sentence: "Ce nombre est défini à la fin du XVIIe siècle, dans une correspondance entre Leibniz et Christian Huygens, comme étant la base du logarithme naturel."  Thus clearly, even in French, this usage is quite standard.  Tito Omburo (talk) 20:59, 17 April 2024 (UTC)
The problem with "the base of the natural logarithm function" is that functions generally do not have a base.  Logarithms, on the other hand, have a base.  For example the natural logarithm is one logarithm, which has base e, and the common logarithm is a different logarithm, which has base 10.  Logarithms, in this sense, are not numbers. --Trovatore (talk) 19:00, 17 April 2024 (UTC)
When you say logarithm you either mean a number or a function. A logarithm (singular) that is a number is a specific instance of the logarithm function. So when you make a statement about 'the natural logarithm' then either you are talking about the function or just one number. If you are talking about a single instance then we must ask, "Which instance and why not all the others?" Consequently the statement that e is the base of the natural logarithm is only meaningful if it refers to the natural logarithm function. OrewaTel (talk) 19:30, 17 April 2024 (UTC)
Basically "the natural logarithm" in isolation always refers to the function, unless there is an understood value to take the logarithm of.   So the sentence "e is the base of the natural logarithm" is clear and unambiguous.  The problem with "e is the base of the natural logarithm function" is that the object of the preposition has "function" as the head, and "the base of a function" makes no sense. --Trovatore (talk) 19:34, 17 April 2024 (UTC)
Agree.  Basically, a logarithm is its own thing.  Numbers don't have bases; functions don't have bases.  Logarithms have bases.  Tito Omburo (talk) 20:56, 17 April 2024 (UTC)

In https://en.wikipedia.org/wiki/Jacob_Bernoulli the year of discovery is stated as 1683 while in this article it is 1685. Can someone clarify the discrepancy. Thanks 192.56.200.12 (talk) 13:34, 18 April 2024 (UTC)

I think Bernoulli's work wasn't published until 1685.  Both articles give 1683 as the discovery date.  Tito Omburo (talk) 15:31, 18 April 2024 (UTC)

This seems quite strange because any number raised to the power one equals itself. What's the point?
"The equation e=e^1 implies therefore that e is the base of the natural logarithm." ★NealMcB★ (talk) 14:10, 5 June 2025 (UTC)

The expression 
  
    
      
        
          e
          
            x
          
        
      
    
    {\displaystyle e^{x}}
  
 is not defined as 
  
    
      
        e
      
    
    {\displaystyle e}
  
 raised to the power 
  
    
      
        x
      
    
    {\displaystyle x}
  
.  It is a mathematical function, the natural exponential function that is defined independently of some other possible definition of exponentiation.  Indeed, raising to a (real) power is defined in terms of the natural exponential function!  Tito Omburo (talk) 18:25, 5 June 2025 (UTC)