Talk:Natural logarithm

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• Talk:Natural logarithm/Archive 1

Which base is the default for "log"?

I'm sure I'm beating to death a dead horse, but I wanted to register my strong DISagreement with the decision to allow "log" to stand for log base 10 by default. I have several reasons for this, which I'll give below, but first I wanted to respond to a strange remark in the article:

At the time of this writing (2003), many mathematicians have adopted the "ln" notation, but "log" also remains in widespread use.

Really?? This statement makes it sounds as if "ln" is the usual convention, and those that "still" use "log" are the last dying members of a minority that hasn't updated itself to common usage. This is anything but true. In fact, in mathematics, "log" _IS_ the standard notation for natural logarithm, and virtually the only time I have ever seen a mathematician use "ln" was in teaching undergraduate calculus or differential equations classes. Let's not give the impression that "ln" is "finally taking over". "log" isn't just in "widespread" use. It _IS_ the standard convention. "ln" is considered non-standard, although recognized, notation. But the most important part is that "log" is ALWAYS assumed to represent natural logarithms in mathematics. If you write something like "log 1000 = 3", that would be considered a false statement. The only reason math people use "ln" is because they feel forced to in calculus and diff eq classes.

Now, here are the reasons I think "ln" should remain an ACCEPTED convention at wikipedia, but why "log" should NOT be taken to be base 10 by default:

1. The "ln" notation is fleeting. The only reasons it still survives are the force of history and the manufacturing design of calculators. With modern computer systems and programs today, however, there is no need at all to consider a special base like "10". We don't use base-10 tables to computer arithmetic by hand. And it's possible to solve ANY log problem without using a specific base, because the logs between different bases are related by a simple formula. (In other words, when someone says "take the log of both sides", you can do base 10 if you want, or base 2, or base e, or base google, or whatever, the problem will solve itself eventually if you keep track of the base.) But as computer systems develop and second- and third-generations enter the teaching force who don't use "ln", the notation will eventually fade away.

2. The natural logarithm has enormous special mathematical significance out of all possible logarithms. Choosing the base "e" is not convention -- there are definite mathematical reasons why "e" really is not the same as other bases and why we want to work with it more than any other base. Given its enormous mathematical significance, far more than "10", the default value of the notation "log" (which is clearly the more obvious notation for a logarithm than "ln") should be the base "e".

3. As I mentioned above, using "ln" is considered nonstandard notation in mathematics and many areas of science. In advanced mathematics journals, it is as extinct as the dodo bird. The more important point, again, however, is that VIRTUALLY ALL MATHEMATICIANS and a great many more scientists and engineers ASSUME that the default notation for "log" is the natural logarithm. Simply put, if you want to write "ln" for yourself, fine, you will be understood. But if you write "log", people will NOT understand you to mean base 10, they will assume it is the natural log. "log" = natural log is the standard convention, assuming that it means log base 10 is simply non-standard notation, and you will be misunderstood if you write this way.

4. This reason is related to number 2 and I think it is a compelling reason that many people forget about. When the natural logarithm is extended to a multiple-valued complex function on C - {0}, virtually no one that I am aware of (and this includes scientists, engineers, textbooks on complex variables, textbooks on diff eqs, "ordinary" non-mathematicians, etc.) writes "ln z" for this multiple-valued function. They often might write "ln r" for the real part of this function, but I cannot recall a single instance where I saw the notation "ln z" used for the complex function, EVERYONE uses "log z". This choice of notation ("log z" for the complex log function, "ln x" for the real log function, and "log x" for the log base 10) poses a number of real theoretical and practical problems, inconsistencies, and contradictions. For instance, when I write "log 1000", if I consider 1000 as a REAL number, and I use the usual wikipedia convention (that is being advocated) then this expression should have the value of "log 1000 = 3". However, 1000 is also a COMPLEX number, as is every real number, and if I consider 1000 as a complex number, and I use the (universally accepted) notation for the complex function, I get that "log 1000 = natural log 1000", in other words, "log 1000 = ln 1000". So, depending upon which interpretation I give 1000, and which notational convention I choose, I get TWO DIFFERENT ANSWERS for "log 1000", namely "3" and "ln 1000", these are not the same.

To repeat, "ln z" is non-standard notation, I'm a bit flabbergasted to actually see it used here!!! To continue to use "ln z" for the complex (natural) logarithm is actually doing a DISservice to people visiting the site, and it's misleading them.

The point is, this reveals a fundamental flaw in the convention that "log" denote log base 10. By choosing "log" to denote log base 10, we get a conflict and contradiction with an ALREADY USED, UNIVERSALLY ADOPTED notational convention. This both defies common sense and creates a great deal of confusion.

5. As a final note, I'd like to say that I often do a lot of reading of textbooks and papers in analytic number theory or at least number theory involving complex function theory (esp. asymptotic estimates, orders of magnitude of arithmetic functions, etc.) and I can tell you that writing "ln x" or "ln n" or even worse "ln ln ln x" instead of "log log log x" strikes ANYONE's eyes who works in these areas as strange, wrong, and a notation that is never used in these fields. By forcing people who write articles in number theory to do this, e.g. using "ln x", you are essentially forcing researchers and practitioners in these fields to go AGAINST a universally adopted notation, and use one that is never used at all in journals, textbooks, or anything. This is not just true for number theory, (although it does look especially ridiculous in log-log-log estimates in analytic number theory) it is true in almost every branch of mathematics. I quote from Eric Weisstein:

Note that while logarithm base 10 is denoted log(x) in this work, on calculators, and in elementary algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use the notation log(x) to mean ln(x), and therefore use log-10(x) to mean the common logarithm. Extreme care is therefore needed when consulting the literature.

(Emphasis added)

I find it strange to read that although Eric freely admits that log(x) = natural logarithm is a UNIFORMLY used convention, he then continues to go against it. Why would you go against a notation that is uniformly used by people who are the primary practitioners of it?? It makes no sense.

I hope I have shown that this is not just a matter of "mathematical snobbery".

DISGRUNTLED WORKING MATHEMATICIAN

Many people still commonly use "log" for log base 10. If some will get confused if log is meant as the natural logarithm, it should not be changed. Dysprosia 06:44, 8 Oct 2003 (UTC)

I have read and practiced a great deal of math. But I use both log and ln. They just have different domains. ln is clearly a real valued C^infinity function on the domain of positive reals, defined from the integral of 1/t, whereas log is a complex valued analytic function (whose real part is given by ln) defined on whatever branch the context is providing as an analytic inverse to the exponential function. These are different definitions that coincide over the positive reals, which in practice makes ln somewhat useless, but it's very convenient for calculus students to have an entirely real variable based definition. They are, of course, both "base e." Any other base is really just multiplying the log function by a constant, so obviously possible it's truly not worth mentioning. — Preceding unsigned comment added by 75.140.4.134 (talk) 06:45, 28 May 2011 (UTC)

Many (british?) textbooks use lg for log₁₀, since this avoids confusion as well. Personally, lg is ugly, and I don't like writing log₁₀, but one of them is necessary. log_c (or a) is also often used to indicate that choice of base in arbitrary. For common use of plain log, in math (and probably most of physics) it means base e, in chemistry and astrophysics it means base 10. Elektron 08:52, 2 May 2004 (UTC) (I was in a rush earlier and forgot to sign it)

Forget it. Obviously this is pointless.

So, if the people who are WORKING in the field use one notation, and the people a number of people outside the field use another, we should just adopt the one the working practitioners don't use because we don't to confuse them? This reminds me of arguments that incorrect grammar should be tolerated because if enough people use them, they should be accepted just because enough people use it. (If enough people write "Their going to play ball", and that's how they spell it, why question?) Aren't they going to be confused when they advance their math skills at wikipedia, and then go out and pick up a book that uses log = natural log and not be able to figure out what's going wrong, why they don't understand? And what about people who use the log = natural log convention (like the numerous academics who visit here) and read an article using "log" and wonder why its not natural log?

Some points: people working in the field use "ln"="log" as well as "log"="log_e". Both ln and log link to logarithm, the general article - both ln and log represent logarithms. Grammatical errors are different from notational differences, which merely express a different way of saying the same thing. Dysprosia 07:13, 8 Oct 2003 (UTC)

Not quite. If this were true, I could say, let "sin" stand for the cosine function, and "cos" stand for the sine function. Now, this is purely a notational difference, that merely expresses a different way of saying the same thing. So, why should I not be able to use "sin" for cosine and "cos" for sine? It's all personal taste.

The point is, it's not all personal taste. If you want to use "ln" for natural logarithm, I may not agree with it, but that's your right. "ln" is an accepted notation for natural logarithm. But you want MORE than that. You want to usurp the notation "log" exclusively for log-base 10. And this is tantamount to making the kind of switch above. You're not just asserting the right to use your own notation (which is fine), you're TAKING AWAY the notation used by most people in the field (and again, even though some people do use "ln", they're pretty rare, at least in a clear minority). The difference is this: EVEN AMONG ACADEMICS AND RESEARCHERS WHO USE "LN" NOTATION, THEY STILL ACCEPT THE DEFAULT NOTATION OF "LOG" AS NATURAL LOG, EVEN IF THEY USE "LN", THEY DON'T ASSUME BASE-10. This will have the following effect: researchers and mathematicians who come to wikipedia will write tons of articles in math and science, and the vast majority of them will go right ahead and use the notation familiar to them, and you will have an enormous pruning job on your hands going around cleaning up after them, and then explaining to everyone what the policy is, etc., etc. Moreover, they won't be aware of the policy until you tell them, or until they accidentally run into it themselves. The vast majority of people will come here writing articles, using "log" (at the very least, if a large number of mathematicians eventually come) and you'll be the one left to do all the pruning and explaining, just don't complain about it when it happens.

No, that's an agreed convention to use "sin" for sine and "cos" for cosine. I personally don't want to usurp "log" for log₁₀, it is Wikipedia convention as mentioned in the article - another agreed convention to use log for log₁₀, as it is on Mathworld as you mentioned.

By the way, I personally won't have to change the names if people add to the work, many will; Wikipedia is collaborative - many hands make light work.

And it's not 'agreed convention' to use log for log₁₀, but most people use it anyway. Let's give an example. Most people measure angles in degrees (mechanics and engineering, I believe). We do this because we're used to dividing a circle into 360 bits. So people get into the habit of saying stuff like "sin 30 = 1/2" (which is incorrect), when they mean "sin 30° = 1/2" (where ° = π/180), unless they forgot to switch the calculator to radians, and then they mean "sin 30 = -0.998...". Both meanings of log are in Common logarithm.

Strangely, logarithm suggests that lg sometimes means log₂, while I've only seen it mean log₁₀ (this is in many of my (British) school textbooks).

I propose we just stick to the convention for the topic the article is written in. PH uses log₁₀ for the first occurrence, then log later. ln or log_e can be used for natural logarithms, depending on style. Elektron 08:52, 2 May 2004 (UTC)

And in mathematics, it's also an AGREED convention that "log" means natural log, that's the point you don't seem to understand. You're telling people that the notation they use in wrong, an AGREED upon notation. "log" = natural log is as agreed a convention in mathematics as "sin" = sine or "cosine" = cosine. Ask virtually any mathematician, if they see the notation "log", what do they ASSUME (without any other information) that it means, I guarantee you, 99% of them will say natural log, unless told otherwise.

You just still don't understand apparently. You're not going to change the convention of the math world, and in effect what you're doing is saying, "if you want to write an article here, you can't use the notation that is used 90-95% of the time". That's like inviting French people to submit articles in French, and then criticizing their use of the French language and telling them that the way French is spoken by 95% of the people is wrong and they must change if they want to submit articles. It's asinine. You might be able to prune and edit the problem, but you're going to drive LOTS of people away -- a lot of people are just going to say "screw it -- they can't tell me that I can't use a notation that's used by 95% of the people I know" and they're just going to LEAVE. Is that what you want? You just can't tell people to go against a convention used by 95% of the people and expect them to just go along -- many will raise objections, the rest will just get pissed off at the whole thing and leave.

log is also convention for log₁₀, so it can't just be changed without confusion. What about the people who come and see log and think it's log₁₀? A compromise would be to have log ≡ ln, and change all the old log to log₁₀, but one idea does not make a consensus. We need other points of view before making any changes. Dysprosia 22:41, 8 Oct 2003 (UTC)

I have some comments along these lines on the talk page for logarithm Talk:Logarithm

My own opinion is that "$\ln$" notation is ugly, hard to read, unintuitive, and completely unnecessary. In languages such as English, written symbols represent sounds, and it is the sounds themselves that carry meaning. When we learn to read, we "sound out" written words and interpret the sounds those words represent. As we get older, we never actually outgrow this process. The brain pathways may become a little more streamlined, but one's reading comprehension remains basically a process of sounding out words and interpreting the meaning of those sounds. This habit tends to carry over into mathematics even though there the written symbols directly carry the syntactic meaning, and there is no standard way to pronounce even a moderately complicated mathematical formula. Witness lower down on this talk page some of the rather humorous idiosyncratic pronunciations of "$\ln$." That alone should be evidence for my claims of "hard to read" and "unintuitive," whereas "$\log$" is intuitive, easy to pronounce, and therefore easy to read in the context of many languages.

The claim of "ugly" on the other hand is admittedly a personal aesthetic judgment, but my feeling on this is so strong that I will often, upon seeing "$\ln$" used in a book, rewrite the formula in question in a notebook in simpler, more elegant, and more beautiful "$\log$" notation, and then slam the book shut in disgust and work the whole concept out myself in notation that pleases me.

While it is true that mathematicians often look down their noses at "$\ln$" notation, it is not at all from mathematical snobbery; it actually comes more from a notion commonly known as KISS (Keep It Simple Stupid). Once we have identified, among all bases that a logarithm could possibly have, one that seems most natural to us, namely Euler's number, it is also quite natural that "$\log$" without further qualification or specification would signify the logarithm to that base, and that if one wanted to use a different base, one would so indicate. Thus it is redundant to write "$\log_e$," and such abbreviations as "$\ln$" or "$\lg$" are totally unnecessary and rather confusing. Just my not-so-anon two cents' worth. 130.94.162.64 09:13, 23 November 2005 (UTC)

Another complaint: What's with the "$+C$" term explicitly added to every single indefinite integral? This practice almost gags me. It's OK when indefinite integrals are first introduced in a basic calculus class, but thereafter it should just be omitted and left implicit. Students should simply be reminded from time to time that an indefinite integral will remain valid when an arbitrary constant is added. There is no reason to completely subsume a letter of our limited alphabet for such a vacuous purpose. I am sick and tired of these perennial calculus teaching fads, "$\ln$" and "$+C$" aong them. There is nothing wrong with the classic notation of Newton, Leibnitz, and their contemporaries. This is all just change for the sake of change, a kind of mathematical newspeak if you will, and may even have the sinister purpose of raising the barrier of entry to higher mathematics. 130.94.162.64 19:48, 25 November 2005 (UTC)

Shouldn't this article redirect to logrithm, since ln is just log_e? Granted, the natural logrithm is very useful, but it is still just a logrithm.

I like the idea that suggests using log_e to represent a logarithm with the base of 2.7… and using log₁₀ to represent a logarithm with the base of 10. It is a very clear method of representing logs of differing bases. Jecowa 06:38, 21 June 2006 (UTC)

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A logrithm? What's that? Is that the faint distant drum you hear when you put your ear real closed to a log? :-) But seriously, the natural logarithm is not "just a logarithm"; it has many special properties that other logarithms don't have and is the source of the definition of e. That warrants a separate article. -- JanHidders

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Could someone who understands this perhaps explain why ln is useful to normal people? For example, I'm involved in a particular sporting endeavor (indoor rower) where if you plot max speed over distance you get a curved line that corresponds very closely to a ln trendline. Just a thought user:Verloren

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"however in that field too then "ln" notation is coming more and more into use. " -- really? it wasn't that long ago I did my degree in maths, and we pure mathematicians had a standing joke about physicists getting their hands dirty with other log bases. I'm sure pure maths snobbery lives on, and using "log" (to assert that other bases are a waste of time) is part of that -- Tarquin

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Well, I did a study in (theoretical) mathematics too, and I remember having been told that log(x) was used in either meaning, but in practice the only things I have seen are (often) ln(x) and (much less often) ^alog(x); as far as I can tell, log(x) is used in neither meaning in theoretical mathematics nowadays. But then, I went on in other areas than analysis, so I am not your ideal source on this either. -- Andre Engels

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The extension of ln z for arbitrary complex numbers z is slightly wrong. One way is to accept ambiguity: the answer is only defined up to a multiple of 2 * pi * i. If a continuous defintion of ln z is desired, it is standard to exclude the negative real axis from the domain of ln z. It turns out that you can exclude any path from the orgin to infinity and still get a well-defined ln z. This is sometimes important in Complex Analysis.

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I disagree strongly. Although, as was stated in the example, 1000 is complex, to represent logx as logz simply because it is defined over a subset if complex numbers makes no sense. The naming convention from which the difference between z and x arises is widely accepted to mean that f(z) implies f is defined over the complex numbers, which logx is not and logz is. Since it is known that log₁₀ is not defined over the complex numbers, it should be quite clear that it could not, by that convention, be represented as logz. On the other hand, I don't see what the point in calling it logz is, when lnz works just as well.He Who Is 02:11, 18 May 2006 (UTC)

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Thanks to Michael Hardy for putting in the bit about usage of ln / log. Even more annoying that seeing "ln" written is people pronouncing it "lun" ... ;-) -- Tarquin 11:07 Jan 21, 2003 (UTC)

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I've just finished a Maths A-level (here in England); we used ln x for natural logarithms, log₁₀x for logs to the base 10, and log x for logs to an unspecified base. And we pronounced ln "lin". Unfortunately, our Physics teacher used "lin graph" as an abbreviation of "linear graph" and "log graph" as an abbreviation of "(natural) logarithmic graph"... --Greg K Nicholson 03:11, 2004 Aug 23 (UTC)

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I want to convert the following equation from the article into TeX, and I'm just wondering why with the integrals (for example the first one "from" 1 "to" ab) the "from" precedes the integral sign? Is there any special significance to this or is it just wrong / an alternate way of writing integrals:

ln(ab) = ₁∫^ab 1/x dx = ₁∫^a 1/x dx + _a∫^ab 1/x dx = ₁∫^a 1/x dx + ₁∫^b 1/x dx = ln(a) + ln(b).

Thanks, snoyes 04:29 Mar 3, 2003 (UTC)

I expect it's a hack to make the ascii version look not-so-ugly -- Tarquin 13:44 Mar 3, 2003 (UTC)

Yip, I suspected that. Thanks, snoyes 14:35 Mar 3, 2003 (UTC)

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I removed this

In simple terms, the natural logarithm function, or, accordingly, powers of e, occur frequently in natural processes (which is why it's called natural logarithm), and, curiously, e to a given power in calculus is its own differential or integral, meaning that it remains constant in calculus.

since we already have a section about "why it's natural" which gives some better reasons.

I also removed

It should be noted that the reason that "natural logarithm" is abbreviated "ln" and not "nl", which is more natural for English speakers, is due to French influence in naming conventions. In French, "natural" follows the noun "logarithm", and this convention has held in much the manner that the International System of Units is abbreviated "SI".

In school we learned it was from the Latin "logarithmus naturalis"; in any event, since ln was invented by an American professor, the influence of the French seems limited. AxelBoldt 05:32 24 Jun 2003 (UTC)

I learned that it was a French influence, and some websites seem to indicate this as well, but I trust that you probably know a lot more about this than I do. Regardless, I think some short explanation about why it's "ln" and not "nl" should be included. -- Minesweeper 07:58 24 Jun 2003 (UTC)

pronunciation

This article nowhere mentions how to pronounce the expression "ln x." In the united states at least there is great variation — people say, variously, "ell-enn" "lin" "log" (even some who always write ln) and "lawn." I have never been able to find a geographic basis for the variation; it seems simply to depend on people's high school trigonometry or precalculous teachers. Doops 17:26, 25 Oct 2004 (UTC)

Also, two bits on the dead horse of whether the natural log should be written "log" or "ln": the proper course is absolutely clear: using the general terminology, familiar to millions of peopel, will confuse professional mathematicians less than using mathematical terminology will confuse non-professionals. (Furthermore, I don't think mathematicians really think of "log" as standing for "natural log" — they think of it as standing for any generalized logarithm, not really caring (since they're doing theoretical work) what the base is. Then, of course, if they come to resolve the logarithm, they just naturally treat the base as e since, other things being equal, that's the "natural" base.) Doops 18:04, 25 Oct 2004 (UTC)

I used think 'lun' (or 'lunn' depending on your pronounciation rules), personally. But defining log = log_e is convenient, since log(1+x) = x + x^2/2 + .... and then, when integrate 1/x to get log x, you don't mean any base (the arbitrary base is usually c, and probably explicit to show that it's arbitrary). Some mathematicians use log and some physicists use ln, just like some physicists don't like differential form, while mathematicians may not care (if you can extend things without breaking anything, who cares?). I'm training myself to write 'log' since 'ln' looks silly. My calculator also has SI prefixes, but no universal constants! Argh! (the idiots who design calculators these days...) --Elektron 22:32, 2005 May 30 (UTC)

It isn't true that "log" doesn't mean any particular base. It means specifically base e, and, for instance, the prime number theorem is only true if the logarithm is to base e. Eric119 16:02, 2 Jun 2005 (UTC)

Personally I consider "ln" just an alternative spelling of "log", and pronounce it "log". --Trovatore 14:33, 9 September 2005 (UTC)

In the midwest, it seems to be said "ell-en" or "natural log". Yes, saying natural log is sometimes easier than arguing over how to say it. ;P -Matt 01:37, 7 May 2006 (UTC)

Why do you need the word "natural"? Is "log" not "natural log" by default, for you? (If it isn't, then I take it your field is something other than mathematics.) --Trovatore 01:41, 7 May 2006 (UTC)

Eric, in the article Rime Number Theorem, the definition uses ln, not log. -- He Who Is^{[ Talk ]} 12:00, 10 July 2006 (UTC)

About pronunciation, there is a limerick (by Eve Andersson) that implies "ln" rhymes with "sin":

 There once was a log named Ln
   Whose life was devoted to sin.
   She came from a tree
 Whose base was shaped like an e.
 She's the most natural log I've seen.

Yrogirg (talk) 19:17, 25 November 2010 (UTC)

History Section

From the perspective of a near total N00B (me), interested in Euler and Napier, I found it interesting that "e" and Euler are mentioned everywhere, but the 800 pound gorilla is not at all, namely that "e is Euler's number" - only a footnote hints at this and, of course the first sentence " the base e, where e is an irrational constant " has a link to base e, but again it's not self evident. thanks - 98.82.79.14 (talk) 18:19, 27 February 2010 (UTC)HP

Using pre-calculated values

This is a copy of a comment I made on the talk page for the mathematics project:

It occurs to me that one could pre-calculate some values of ln to high precision and use them in a subroutine which calculates the natural logarithm. For example, pre-calculate ln(2), ln(3/2), ln(5/4), ln(9/8), ln(17/16), ln(33/32), ln(65/64), and ln(129/128). Suppose we are given a number x as input. Initialize the output to zero. Shift x into the range from 1/2 to 1 while adding (subtracting) ln(2) to the output for each bit which x was shifted downward (upward). Then if x<2/3, multiply x by 3/2 (just a shift and add) and subtract ln(3/2) from the output. If x<4/5, multiply x by 5/4 (just a shift and add) and subtract ln(5/4) from the output. Similarly for 8/9, 16/17, 32/33, 64/65, and 128/129. Then 128/129≤x≤1 is close enough to 1 that the power series for ln(x) should converge rapidly. Add the (negative) series sum to the output and return that.

JRSpriggs (talk) 10:49, 11 November 2010 (UTC)

The J-ln and Eulers triangle

An approximation to the natural logarithm that remains near the natural logarithm over a greater domain than any known series approximation of the natural logarithm is the J-ln (Johnson's natural logarithm). This recently developed approximation was derived by taking the derivative of an antiderivative algorithm for exponential functions of the form $b^x$. The result of this derivation are rational expressions that appear to converge to the natural logarithm as the index of the iteration of the original antiderivative algorithm increases. Specifically, the numerator and denominator polynomials that compose this rational expression have coefficients that can be replicated through the numbers contained in successive rows of Euler's triangle. Consequently, due to the connection with Euler's triangle, the J-ln approximation to the natural logarithm can be expressed as

$\ln(x) \approx jln_n ({x}) = \frac{n (x-1) \sum_{t=0}^{n-2} (x^t \sum_{u=0}^t ({-1}^u) \binom{n}{u} (t+1-u)^{n-1})} {\sum_{v=0}^{n-1} (x^v \sum_{w=0}^v ({-1}^w) \binom{n+1}{w} (v+1-w)^n)}$ for $n>1$.

This relationship has yet to be fully proven, but the J-ln has been shown to produce one significant figure of accuracy for x-values as large as 250,000 at n-values as low as 36. By comparison, the Euler transform of the Mercator series and the complex series approximation to the natural logarithm both require over 2,000 terms to attain a similar level of accuracy for an input value of 250,000. The J-ln is significant because it requires fewer mathematic tricks to produce relatively accurate results than is the case with most series approximations of the natural logarithm.

I would like to see the above section included in the natural logarithm article because it is a significant development in the understanding of natural logarithms (and logarithms in general). It has been previously discussed that adding this section may conflict with Wikipedia's no original research policy, but the formula given above is fairly easy to derive by anyone with appropriate mathematic skills and thus can be presented via very old sources (it does not require that anything new about math be understood). Another concern that was raised was that this formula, if it does not violate the no original research condition, may be too tangential to include in the article. I admit that this formula is not common, but the same could be argued of the Euler transform of the Mercator series in that most ordinary people are not going to know anything about it, yet the Euler transform was included in the article. Can anyone think of other reasons why this section should not be included in the article? I think a large-domain approximation of the natural logarithm is a significant item for an encyclodepic article on natural logarithms to possess. What are your thoughts/concerns about adding this section?Maonaqua (talk) 14:00, 12 May 2011 (UTC)

Even if this approximation is correct and is as simple to derive as you say it is, you still need to provide a reliable source that uses the term "Johnson's natural logarithm" to describe it. Gandalf61 (talk) 14:25, 12 May 2011 (UTC)

Okay, what if I changed the section to simply "an approximation" instead of calling it "Johnson's natural logarithm." It is hard to find a journal article for this formula because it is generally not used too often given the advent of computer algorithms. That is, this is a method that was designed more for hand computations than for computer programming and thus does not see much coverage in academic journals.Maonaqua (talk) 18:50, 12 May 2011 (UTC)

So why did you call it "Johnson's natural logarithm" in the first place if you don't have a source ? I am afraid this is looking more and more like the results of unpublished original research. Gandalf61 (talk) 11:02, 13 May 2011 (UTC)

Just trying to give credit to the guy who first derived it (as far as I can tell), but the result still stands without his research. That is why I proposed dropping the title--the formula is independent of its original derivation in that most mathematicians should be able to derive it. If you want to see how the formula was originally derived, you can check out Johnson's website: http://sites.google.com/site/visualmathorg/ . It should be under the link "Antiderivative Approximation." You will see that deriving the above formula is pretty simple and, if taken as an approximation (hence the $\approx$ sign), is something that most mathematicians would agree approximates the natural logarithm function. (See pages 22-25 and page 36 of his informal article.)Maonaqua (talk) 14:39, 13 May 2011 (UTC)

How it is actually done normally is to calculate the log base 2 using a rational function (one polynomial divided by another) after scaling to between 1 and 2 and then multiplying the result by log_e 2. Works in no time flat and very accurate. There's even versions which return correctly rounded results for all representable floats or doubles with a very respectable speed. Dmcq (talk) 15:40, 12 May 2011 (UTC)

True, but those methods require scaling and predetermined constants (or another method to derive the needed constants). The J-ln is fairly unique in that it does not require thos kinds of mathematic tricks. It is a single formula that can be used to estimate the natural logarithm of almost any number and is self-contained. I am not saying that it is the quickest method out there, just one of the few that is practical to use by itself for most input values.Maonaqua (talk) 18:50, 12 May 2011 (UTC)

P.S., The formula as given is rational--just FYI--and thus can be used in a manner similar to the one you described: a number can be rescaled, computed, and then scaled back to the appropriate value fairly quickly. This formula is different in that rescaling, etc, is not as much of a necessity with this formula. Not necessarily quicker, just more self-contained.Maonaqua (talk) 18:57, 12 May 2011 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘The article already makes it clear how one can compute your example value

$\ln(250000) = 6 \ln(5) + 4 \ln(2) \,$

without any great difficulty. Or see the section of talk above Talk:Natural logarithm#Using pre-calculated values. JRSpriggs (talk) 00:00, 13 May 2011 (UTC)

Granted, but the point is that the J-ln does not require those sorts of modifications--that is what makes it unique. Yes, of course there are both faster and more precise methods out there, but they are not self-contained like the J-ln is.Maonaqua (talk) 14:39, 13 May 2011 (UTC)

Part of the problem that I feel is that the formula looks too complicated. It might help to rewrite it as follows:

$\operatorname{jln}_n ({x}) = \frac{n (x-1) \sum_{t=0}^{n-2} x^t c_{n-1, t}} {\sum_{v=0}^{n-1} x^v c_{n, v}}$

for n > 1 where

$c_{n, v} = \sum_{w=0}^v ({-1})^w \binom{n+1}{w} (v+1-w)^n \,.$

Then the values of the coefficients c_n,v could be tabulated in advance to make the calculations easier. JRSpriggs (talk) 18:01, 13 May 2011 (UTC)

Yeah, that would make it a tad more user-friendly; I'm okay with restating it that way. 134.50.221.21 (talk) 00:57, 14 May 2011 (UTC)

If this formula is put into the article, where would be the best place to put it? —Preceding unsigned comment added by 134.50.221.21 (talk) 00:59, 14 May 2011 (UTC)

Unfortunately, we still need a reliable source which says that this is a good approximation to the natural logarithm. JRSpriggs (talk) 06:56, 14 May 2011 (UTC)