Talk:Natural logarithm

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For two reasons, I question the statement:

"Indeed, Nicholas Mercator first described them as log naturalis before calculus was even conceived."

First, the work in which Mercator apparently used this term was published in 1668, whereas Newton had invented calculus around 1666.

Second, I think Mercator was not the first to use the term "natural logarithm." In A History of Mathematics (1968), Carl B. Boyer writes, "Mercator took over from Mengoli the name 'natural logarithms' for values that are obtained by means of this series." This suggests Pietro Mengoli coined the term, or at least used it before Mercator.

I did not do any editing of the actual article.

Jeff560 04:53, 10 July 2006 (UTC)




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)
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. logc (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"="loge". 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 loge 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 "loge," 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 loge? Granted, the natural logrithm is very useful, but it is still just a logrithm.

I like the idea that suggests using loge 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)

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

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

"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

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

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.

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)

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)

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)

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)

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)


________________________________________________________________________________

Around 1975, Felix A. Keller of Switzerland discovered the following formula that converges in e:

lim→ ∞ {(n^n)/[(n-1)^(n-1)]} - {[(n-1)^(n-1)]/[(n-2)^(n-2)]} for n>2

This formula was published for the first time 1998 on Steven Finch's website www.mathsoft.com/asolve/constant/e/e.html. He commented: “This is a pretty formula! It is fun to generalize things like this: I haven't seen limits like this before.” He refers to it as “Keller's Expression”.

Pretty, but hardly a surprise since (n^n)/(n-1)^(n-1) gets closer and closer to e*(n - 1/2).--Henrygb 16:51, 29 Sep 2004 (UTC)


This is an apology for my stubbornness regarding the formula concerning the deriving of pi from ln(ln(ln(ln(e+x))))=a+pi*i, where x>0 and a is a meaningless value. I had discovered by myself that it worked with 10, thought myself clever, and failed to realize the fact that this was merely a special case in which x=(10-e). I also did not see the similarity in the corrected formula because I originally failed to grasp that Im(x) meant the imaginary coefficent of complex number x. Just something I wanted to say. Thanks.

No problem. Don't worry about it. Dysprosia 08:26, 23 Apr 2004 (UTC)

[edit] possible logarithm bases

The article on the natural logarithm says that the base of a logaritm can be any positive number greater than 1. It should say, "other than 1". -Mike Jones

Fixed. Dysprosia 01:35, 2 May 2004 (UTC)

[edit] 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)

[edit] Alternative way of calculating ln

\ln(1+x)= x \,( \frac{1}{1} - x\,(\frac{1}{2} - x \,(\frac{1}{3} - x \,(\frac{1}{4} - x \,(\frac{1}{5}- \ldots ))))) \quad \left|x\right|<1.

As far as I can see (and I'm too lazy to make sure) this is the same as math2.org's \log (x+1) = \sum_{n=1}^\infty {x^n \over n}. I think the symbolic form is more legible, but more importantly, math2.org claims it is valid for x == 1, while the above excludes x==1 from the domain. --대조 | Talk 12:47, 8 April 2006 (UTC)

Both expressions give you the same sequence of approximations, so they both converge to the same value (log 2) when x=1. But it's what's called a conditionally convergent series, meaning that the sum depends on the order in which you add up the terms. While it's easy to see that the above series converges (by the alternating series test), it's not trivial to see that it adds up to the "right" value, log 2.
It's also a seriously inefficient way to compute log 2. After a hundred terms you'd still be working on the third significant figure. By contrast, if you give fifty terms each to the expressions for log(3/2) and log(4/3) and add the results together, you should get about 16 digits right, at a back-of-the-eyelids estimate. --Trovatore 19:11, 8 April 2006 (UTC)
Oh, one more point—there's an error in your infinite sum. Should be \log (x+1) = \sum_{n=1}^\infty {(-1)^{n+1}x^n \over n}. --Trovatore 19:30, 8 April 2006 (UTC)

[edit] proposed merge about notation

I think merging the discussion about notation with the other material would be a mistake, at least, if it ends up leaving this article. People need to read this about notation -- o/w they will read things wrong. Revolver 03:52, 18 August 2005 (UTC)

[edit] definition of lnx

is there any other way of defining lnx not using integrals? like i know no algebraic way but what about limits? dont laugh but i really havent learned integrals yet (hate them, they are everywhere)

—The preceding unsigned comment was added by Protector (talk • contribs) at 11:19, 6 September 2005.

If you know the exponential function ex, then you can define ln as the inverse of this function: if ex = y then ln y = x. Alternatively, you can use the Taylor series for x ∈ (0,2)
\ln (1+x) = x - \frac12 x^2 + \frac13 x^3 - \frac14 x^4 + \frac15 x^5 + \cdots;
for x ≥ 2, you use the rule
\ln (2x) = \ln x + \ln 2. \,
I hope that answers your question. By the way, it helps if you sign your comments on the talk pages; you can do that by typing four tildes, like this: ~~~~ . -- Jitse Niesen (talk) 13:02, 6 September 2005 (UTC)

[edit] programming languages

I've added a bullet on programming languages to the Natural_logarithm#Notational_conventions section. It's from memory; someone please check it for accuracy (and whether Pascal should be included). Any corrections should also be applied to the corresponding bullet at Logarithm#Unspecified_bases. Thanks, Trovatore 05:23, 9 September 2005 (UTC)

In Microsoft Excel XP, LOG means log10 (at least in the Italian version, names of function are different). --Army1987 10:30, 9 September 2005 (UTC)
Looks like you're right. Well, isn't that special. One more thing to hold against Microsoft, I guess. --Trovatore 14:06, 9 September 2005 (UTC)
Excel worksheet formulae use a calculator metaphor. VB for Excel still uses "Log" to mean Log.e(x)

[edit] request for explanation

can we provide some proof as to why you can only take lns of real and complex numbers. a mate of mine showed me this thing that by taking lns of e^i.pi = -1 you get 2i.pi=0 which is nonsense but only because as i learnt later you cant take lns of imaginary numbers. it does imply that here but perhaps we should provide proof lest others slip up the way i did, and, although common sense may dictate that its correct, cater for people like me who dont have much of that. i have one way of prooving it but its shocking, someone with more experience could do better, i wont put it on, but i spent some time thinking on it and i like wikipedia and have no life so here goes since i^4 = 1 if we can take lns 4ln i = 1 (or ln e = 1) ln i= 1/4 e^1/4= i which is nonsense

Your first error is in the second step, as you fail to take logs of both sides there, and you cannot rely on ln ab = b ln a for a,b complex. Dysprosia 15:05, 5 January 2006 (UTC)

[edit] Recursive confusion

Okay, this problem has been irking me for quite some time now. We have two definitions of what the natural log function is:

\ln x = \int_{1}^{x} {1 \over t} dt (integral definition)
ln(expx) = exp(lnx) = x (inverse functions)

Now, one can prove that {d \over dx} \ln x = {1 \over x} using the inverse definition, the fact that {d \over dx} \exp x = \exp x (which can be proved using implicit differentiation and the fact that what we're trying to prove is correct*), and the formula {d \over dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}, but as you can see, we've just went around in a circle trying to prove one or the other. Basically, unless someone can explain otherwise, ln(x) can only be defined via one method, and that method can be used to prove the other method (e.g. using the integral definition and some other theorems to prove that ln(exp(x)) = x).

  • proof:
y = ex
lny = x
{d \over dx} (\ln y) = 1
{y' \over y} = 1
y' = y = ex

As you can see, this depends on both the integral definition and inverse function definition of the natural log. I'm assuming that the inverse relationship of ln x with exp x is either a postulate (which would suck badly) or a theorem. I've tried to prove the derivative of both exp x and ln x using the limit definition:

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

but I've been unable to prove either limit exists without using L'Hôpital's Rule (which would be pointless to use in this circumstance anyhow).

I've thought of using the power series definition of exp x: e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}, but that also would assume that the derivative of exp x = exp x. So, power/Taylor series are out of the question as well.

So, what's wrong with math? I'm sure the recursion has to end somewhere... -Matt 02:02, 7 May 2006 (UTC)

[edit] ============== Compound interest =============

I'm not a mathemetician, employed or otherwise. But shouldn't a discussion of "e" meant for the masses like me include the good old formula (1 + 1/n)^n, n --> infinity?

This business about the "natural" part of "natural log" being due to its widespread utility is nonsense, right? Isn't it due to that basic formula?

RandyT5194 14:04, 19 May 2006 (UTC) A regular human from North Carolina

I, too, am not a mathematician, (Just a kid who is fed up with the general assumption that it seems is made by any system of formal education that any person's knowledge is limited by their age, and that a grade acts as any reflection thereof) but I believe you may be slightly mistaken. The formula you mentioned, while it is the formula orginally used to find e, is not the reason for ln x being "natural." The article contains a section on it, but a few of the primary readons includde that that for the function logbx, the slope at any point equals 1/(x+ln b), and that ln a equals the integral of 1/x, from 1 to a, dx. But the name was originally derived from the natural exponential, e^x because it is its own derivitive. (And is the only funtion with this property.) He Who Is 01:20, 20 May 2006 (UTC)

That basic formula is a corollary of the fact that lnex = elnx = x and L'Hôpital's Rule. Watch:

e^y = \lim_{x \to \infty} (1 + x^{-1})^x

y = \lim_{x \to \infty} x \ln (1 + x^{-1})

Move x to the bottom:

y = \lim_{x \to \infty} \frac{\ln (1 + x^{-1})}{x^{-1}}

Using L'Hôpital's Rule (the above goes to 0/0, so we can use the rule):

y =\lim_{x \to \infty} \frac{ \frac{x^{-2}}{1 + x^{-1}} }{ x^{-2} } = \lim_{x \to \infty} \frac{1}{1 + x^{-1}} = 1

Plugging y back in, we get:

ey = e1 = e

Q.E.D.

(why Wikipedia's TeX rendering engine doesn't support \dfrac{}{} I have no idea, but you'll have to live with the tiny fraction)

-Matt 05:25, 20 May 2006 (UTC)

-Kyle 18:17, 20 October 2006 (UTC)

Just a question, why do we label the natural log as "ln" instead of "nl?" Would it not make any more sense to label the natural log "Natural Log"? An answer to this can be sent to kyle.kruchok@gmail.com

___________________________________________________________________________________

there's another way of calculating e...

e= 2 + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + 1/8!.....and so on to infinity

Greetings,

Nicholas

[edit] Subsection on high precision calculation of the natural logarithm

In this section, it gave a formula for the natural logarithm in terms of the arithmetic-geometric mean (which I never heard of before). I think that some justification is needed for that.

Also I have doubts about the statement that we should "use Newton's method to invert the exponential function, whose series converges more quickly". If one has a value yn near the true natural logarithm of x, then Newton's method yields:

y_{n+1} = y_n - \frac{\exp (y_n) - x}{\exp (y_n)} = y_n + (\frac{x}{\exp (y_n)} - 1).

If we call the increment wn, we get:

w_n = \frac{x}{\exp (y_n)} - 1
y_{n+1} = y_n + w_n. \!

However,

\ln (x) = \ln (\exp (y_n) \cdot \frac{x}{\exp (y_n)}) = \ln (\exp (y_n)) + \ln (\frac{x}{\exp (y_n)}) = y_n + \ln (w_n + 1)
= y_n + \sum_{k=1}^{\infty} \frac{-{(-1)}^k}{k} w_n^k .

I do not see how this could be inferior to continuing with Newton's method since the calculation of w which is the hard part is already done. JRSpriggs 11:53, 21 November 2006 (UTC)

That is a nice trick but it requires a full precision evaluation of the exponential function just like Newton's method.
The arithmetic-geometric mean algorithm for the natural logarithm is described in Jonathan Borwein & Peter Borwein: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity and others. Or more practically, see log.c in the MPFR source code which uses the exact formula given in the article to calculate logarithms.
Jonathan Borwein and David Bailey write in Mathematics by Experiment on pages 227-228 that: For moderate levels of precision (up to roughly 1,000 decimal digits), et may be calculated using the following modification of the Taylor's series ... Given this scheme for et, moderate-precision natural logarithms ... may then be calculated using the Newton iteration ... For very high precision (beyond several hundred decimal digits), it is better to calculate logarithms by means of a quadratically convergent algorithm due to Salamin [the AGM] ... Given this scheme for logarithms, very high-precision exponentials can be calculated using Newton iterations .... Fredrik Johansson 15:33, 21 November 2006 (UTC)
I suggest that you put that reference into the article in place of the "citation needed" tag.
In response to your statement that "... but it requires a full precision evaluation of the exponential function just like Newton's method.": Yes, one has to calculate the exponential once (assuming a well-choosen starting value, say y0 correct within 10-8 as one might get from a hand-held calculator) and do a division with either method. However, if one were using Newton's method, one would have to repeat the process at least one more time to acheive a precision as good or better than that of the method I described. So the choice is between doing an exponential series applied to a number (y1) which may be far from zero (and thus converges slowly) or applying the natural logarithm series to a number (w0) which is presumptively quite close to zero (and thus converges quite rapidly). [Notice I made some fixes to my previous message.] JRSpriggs 05:52, 23 November 2006 (UTC)

[edit] Natural Log formula

I was doing my math homework one night while making a program to do it for me using TrueBasic Bronze software, and I need it to calculate the Natural Log (ln) of a number, since TBbronze doesn't do that, is there a formula to find the natural log of a number other than inputting it into my TI-83 Plus?Drag0nslayer4G0d 00:44, 22 November 2006 (UTC)DS4G

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