Talk:Natural logarithm/Archive 1

Origin of the name "natural logarithm"

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)

Some limits

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)

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)

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)

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)

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 e^x, then you can define ln as the inverse of this function: if e^x = y then ln y = x. Alternatively, you can use the Taylor series for x ∈ (0,2)

$\ln(1+x)=x-{\frac {1}{2}}x^{2}+{\frac {1}{3}}x^{3}-{\frac {1}{4}}x^{4}+{\frac {1}{5}}x^{5}+\cdots ;$

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)

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)

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 a^b = b ln a for a,b complex. Dysprosia 15:05, 5 January 2006 (UTC)

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)

$\ln(\exp x)=\exp(\ln x)=x$ (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:

$y=e^{x}$

$\ln y=x$

${d \over dx}(\ln y)=1$

${y' \over y}=1$

$y'=y=e^{x}$

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)

Compound interest

I'm not a mathematician, 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 log_bx, 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 $\ln e^{x}=e^{\ln x}=x$ 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:

$e^{y}=e^{1}=e$

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)

OK, now that I'm in a discrete math course, I know why this proof doesn't make sense due to its recursiveness: it begs the question, and thus is not a valid proof. I still don't know the answer to this problem, and being away from calculus for so long has made me forget the actual problem I had with this, but I can see the annoyance with the question begging going on here. -Matt 01:12, 22 March 2007 (UTC)

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

Just because you can go down a road for a ways, make a U-turn, and come back where you started from, does not show that the road itself goes around in circles. Why don't you pick one definition, doesn't matter which, of either the exponential function or the log function, and then see if there's anywhere you can't get to non-circularly from that definition. --Trovatore 03:07, 22 March 2007 (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

I think it's because of how you say "natural logarithm" in latin. -Matt 01:12, 22 March 2007 (UTC)

___________________________________________________________________________________

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

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 y_n 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 w_n, 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), e^t may be calculated using the following modification of the Taylor's series ... Given this scheme for e^t, 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)

If you have a good way to calculate the exponential function without the logarithm, you can just do $\ln(x)={\frac {x^{a}-1}{a}}$ and set $a=10^{-32}$ or something else very small. --Thomasda (talk) 15:48, 18 June 2009 (UTC)

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

If the language (which I am unfamiliar with, sorry) has a way to do variable base logarithms, you can use a log base e. Sorry if this doesn't help you. -Tombrend (talk) 19:52, 21 April 2008 (UTC)

I bet you have turned in your homework by now; LOG(X) is TB for ln(x) but in general, log(any base)(x) = log(some base)(x)/log(some base)(any base). Sdegnan (talk) 16:11, 16 July 2010 (UTC)

ln vs log

I was just passing by looking for inspiration on how to improve the Danish article regarding the same subject. I began reading the article but got confused when I reached the Definitions section, because here the notation switches from ln to log.

I was confused by the apparent inconsistent notation.

Then I thought: I better look at the talk page - and wow, what a bag of worms! I see that historically there has been a most heated and long debate on the use of ln vs. log. A debate about feelings, conventions within different fields etc. As I understand the discussion many mathematicians are compelled by the ln notation, because for a mathematician log always means the natural logarithm (of course). I understand that. However, I think that for many others the views can be different. Personally, I'm just a physicist having a perhaps more relaxed attitude towards these puristic views.

My view on this is: Why not use ln consistently as the notation for the natural logarithm? It may not look nice for a mathematician, but the good thing about that notation is that is it unambigious. Although some may not like it it has one great advantage; there is no doubt at all what the base number is! It is e! The mathematicians will say: Ugly and unnecessary notation and of course the base is 'e' - it always is. The physicists will say: How nice - exactly the same notation as in all my text books and articles. Some engineers would wake up: What it is not base 10?!? Therefore, I suggest, with the risk of being flamed: use ln consistently as notation for the natural logarithm in this article!

Also, considering that readers can be curious high school kids who only knows the logarith from the log and ln buttons on their calculators, using a consistent ln notation could avoid quite some confusion for newcomers to this field.Slaunger 22:48, 4 January 2007 (UTC)

Good point. I changed all the "log"s which represent natural logarithms to "ln"s. JRSpriggs 08:49, 5 January 2007 (UTC)

Thank you, JRSprings! To my mind it is much clearer now.Slaunger 10:29, 5 January 2007 (UTC)

Another reason for why the natural logarithm is 'natural'

For every a,
$\ a^{0}=1$
But only e has this property:
$\lim _{x\rightarrow 0}e^{x}=1+x$
So log(1+x) does not approach zero with x in the same way ln(1+x) does, in other words ln(1+x) approaches 0 in exactly the same rate in wich x approaches 0. —The preceding unsigned comment was added by Protious (talk • contribs) 12:06, 24 February 2007 (UTC).

I think you mean $\lim _{x\rightarrow 0}{\frac {e^{x}-1}{x}}=1.$ JRSpriggs 12:17, 24 February 2007 (UTC)

Hyperbolic Log

I am no expert on the subject, but when I mentioned this article in class: my calculus teacher said that a natural log and "hyperbolic logarithm" as mentioned in the first paragraph are NOT one in the same. —Preceding unsigned comment added by U235 (talk • contribs)

Ask him what it does mean, and let us know. JRSpriggs 06:10, 5 June 2007 (UTC)

Searching for "hyperbolic logarithm" on MathWorld gives natural log. as the first result [1], and a Humboldt State University page [2] says that "In 1676 Isaac Newton wrote a letter ... the natural logarithm which was also called the hyperbolic logarithm in those days." The usage may well be very archaic, so your teacher may be understandably unaware. -- atropos235 ✄ (blah blah, my past) 02:52, 7 June 2007 (UTC)

Reason for being "natural"

Isn't the base 10 system totally arbitrary (aside from counting on our hands)? There should not even be a consideration to calling log10(x) the "natural" logarithm, any more than log7(x)... -- atropos235 ✄ (blah blah, my past) 02:43, 7 June 2007 (UTC)

The natural logarithm is base e, not base ten. This is made very clear in the article. -Tombrend (talk) 19:53, 21 April 2008 (UTC)

Bad definition

The statement

the natural logarithm of a number x is the power to which e would have to be raised to equal x

is not a good definition.

This is like saying, "The square root of a number x is the value that, when raised to the power of 2 is equal to x."

What?!? That's like saying, "The word 'good' means 'not bad'".

Zgozvrm 21:46, 24 August 2007 (UTC)

And the problem is... what? Eric119 03:39, 26 August 2007 (UTC)

It's wrapping up the definition of a log with a natural log. Natural log is a log with a base of e. Also, the examples that immediately follow are bad. Any number N for which you take a log base N will be one. Log base anything of one is zero. This doesn't help people understand natural log. It would be more useful helping people understand the concept of a logarithm, which is prerequisite knowledge to this article. 69.38.209.30 (talk) 22:12, 8 February 2008 (UTC)

There should be some mention that the logarithm function (whether "natural" or not) is the inverse function to the power function, just as division is the inverse function to multiplication, or addition is the inverse function to subtraction. —Preceding unsigned comment added by Zgozvrm (talk • contribs) 21:23, 11 June 2009 (UTC)

Natural logarithms and natural history

In the revision as of 06:35, 16 September 2007 I had written under natural logarithm#Why it is called "natural" that, though "common," the base-10 logs are, from a mathematical perspective, rather an arbitrary lot to highlight. My words were: "the only thing special about 10, after all, is the evolutionary accident that it happens to be the number of fingers with which most humans are born." On 22 September an anonymous editor deleted the words "the evolutionary accident," and left the edit summary: "Removed biast towards evolutionary views." Since mathematics remains a science, I reverted that edit. And now a (different??) anonymous editor deleted them again, this time leaving an ungrammatical result.

Now, I have no desire to turn this article into Scopes Trial Redux, but neither do I feel it appropriate to kowtow to the phenomenon—almost entirely American—of willful denial of one of the central tenets of all scientific knowledge. Seems to me that for those who consider the mere mention of evolution prima facie evidence of "biast" [sic], the three-word deletion leaves the remaining argument fairly unconvincing: if you can't square evolution with your understanding of religious doctrine, then the fact of our ten-fingeredness is likely to strike you as divine design rather than the contingent source for various cultural conventions about counting. And in that case, 7 is a superb base for logarithms because it numbers the days of creation in Genesis. But then, so is 4, because the puranas teach that Vishnu must be depicted as a tetrabrachius.

Perhaps there's a way to step back from the brink of culture war. To honor NPOV, I seek others' perspectives.—PaulTanenbaum 03:10, 26 September 2007 (UTC)

Maybe Atropos235 has the right idea: just delete any mention of the common logarithms at all in the discussion of why base-e logs are called "natural."—PaulTanenbaum 03:16, 26 September 2007 (UTC)

Source

Some of the discussion on 'why natural?' needs sourcing. The whole thing about the 10 fingers is OR without a source. Brusegadi (talk) 03:33, 22 December 2007 (UTC)

You had but to ask.—PaulTanenbaum (talk) 02:58, 17 January 2008 (UTC)

Natural phenomena

In the Why it is called "natural" section, the article says "log_e is also a “natural” log due to the large number of natural phenomena that decay at ever slower rates in ways that are described mathematically as simple exponential functions of e". This is incorrect. Yes, there are many natural phenomena that follow exponential decay laws - but there is no "natural" exponent for such laws - the decay curve is exactly the same, whether you describe it using an exponent of e or 2 or 10 or whatever you choose. The mean lifetime parameter is no more "natural" than the half-life - they are related by a simple ratio of ln(2). There are good sound mathematical reasons why base e is the "natural" base for logarithms, but they have nothing to do with exponential decay laws. I propose removing this sentence and the two examples (radioisotopes and temperature sensor) that follow it. Gandalf61 (talk) 14:33, 22 January 2008 (UTC)

Right. Unfortunately, virtually all of the material added by Greg L (talk · contribs) was inappropriate for this section because it did not support the special character of natural logarithms which was the purported reason for adding it. So I removed it. JRSpriggs (talk) 04:05, 23 January 2008 (UTC)

No need to respond JR, I noticed this after leaving you the message. Greg L (my talk) 05:00, 23 January 2008 (UTC)

None the less, let me elaborate a little. Just as y=ln(x) is inverse to x=e^y, so y=log₁₀x is inverse to x=10^y. Thus, this does not indicate that the natural logarithm is special among logarithms, unless you can show that e as a base is special. Your argument could be changed by multiplying all the rate constants by a constant conversion factor of log₁₀e and it would equally well (that is not at all) support the specialness of 10. JRSpriggs (talk) 05:09, 23 January 2008 (UTC)

Good changes (and overdue!), IMHO—PaulTanenbaum (talk) 13:28, 28 January 2008 (UTC)

Mnemonic For Value of e

Here's an easy(?) way to remember the value of e. "Jacksom, Jackson, wore a pair of 45's until he was 90."

Fist, you have to remember that the value of e begins with 2.7. Andrew Jackson was elected president in 1828, so now you have 2.718281828. Then, he wore a pair of 45's until he was 90 sounds like the next digits should be 454590. But Jackson wouldn't wear both 45's on the same hip, thus 459045. Put them all together and you've got 2.718281828459045.

Enjoy,

Larry Gerhardt LarryGrhardt@CS.Com —Preceding unsigned comment added by 74.213.23.26 (talk) 19:08, 5 March 2008 (UTC)

Number of digits of e

Please see WT:MSM#Long numbers for a discussion of the proper format. The majority of mathematical articles, here and in the real world, use 5-digit grouping. — Arthur Rubin (talk) 22:52, 7 October 2008 (UTC)

Almost to a mega-e articles

We are less than 5700 articles away from our 2,718,281st and 2,718,282nd articles. davidwr/(talk)/(contribs)/(e-mail) 05:15, 25 January 2009 (UTC)

Slimming down the introduction

I think there's too much unnecessary information in the introduction of the article. It may scare people away. We shouldn't throw "hyperbolic/Napierian logarithm" or group theory at them in the first paragraph (move to a another/new section?).

Also there is some redundancy in defining ln x through e^x...

What do you think? -- Asakura (talk) 20:01, 2 February 2009 (UTC)

After some studying of my own I realized hyperbolic/napierian logarithm should be mentioned. will fix it -- Asakura (talk) 16:53, 3 February 2009 (UTC)

In the intro, what is t in the formula y = 1/t? Maybe I should know, but I don't, so maybe it is not clear to others! JacquesDelaguerre (talk) 16:46, 7 October 2009 (UTC)

The sentence "The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x." is the verbal equivalent of

$\ln(x)=\int _{1}^{x}{\frac {1}{t}}dt\,$

in which t is the variable of integration. We are trying to avoid confusing t with x which is not really the same thing. JRSpriggs (talk) 09:29, 8 October 2009 (UTC)

JR, maybe since you do that so lovely you should edit it into the intro so it's clearer! I would probably mess it up if I tried. JacquesDelaguerre (talk) 18:17, 8 October 2009 (UTC)

missing information

Whatever information in this article discusses practical consequences of natural logs, it is hidden in the jargon. Please improve this article by providing such information in a separate section. The eigenvalue, eigenvector and eigenspace article is an excellent example of what I'm talking about. 4.249.3.202 (talk) 20:04, 2 July 2009 (UTC)

Logarithms used to be used to reduce multiplication and division to addition and subtraction, but that practical value has been superseded by pocket calculators. Simply, logarithms are important because they arise unavoidably in very many mathematical formulas, especially integrals. JRSpriggs (talk) 06:24, 3 July 2009 (UTC)

Natural Logarithm of a Negative Number

I was interested on what the Natural logarithm of a negative number was. I found out it does have a value. Just that the value is imaginary: ln(-1)=πi Considering the above is true that means that Natural logarithms of negative numbers can be calculated based on this occurrence: Property: ln(ab)=ln(a)+ln(b) Occurrence: ln(-1)=πi ln(-2)= ln(2*-1)= ln(2)+ln(-1)= ln(2)+πi This means that my theory is: ln(-x)= ln(x*-1)= ln(x)+ln(-1) = ln(x)+πi

Now has this concept I believe is original because I have never found anything explaining the negative logarithms. Johnyg200 (talk) 21:48, 28 October 2009 (UTC)

Perhaps something like that should be here, but the natural logarithm in that sense is multivalued; as

$e^{2\pi i}=1,$

It follows that

$\ln \left(-1\right)=\pi i+2n\pi i,$

where n is an arbitrary integer. — Arthur Rubin (talk) 22:31, 28 October 2009 (UTC)

To Johnyg200: Sorry, but that is far from original. See Complex logarithm. JRSpriggs (talk) 09:55, 29 October 2009 (UTC)

Derivation of ln(ab) = ln(a) + ln(b)

According to the article, $t={\tfrac {x}{a}}$ implies:

$\int _{a}^{ab}{\frac {1}{x}}\;dx=\int _{1}^{b}{\frac {1}{t}}\;dt\,.$

Please explain in more detail. -- anonymous 22:21, 2 November 2009 (UTC)

$\int _{a}^{ab}{\frac {dx}{x}}=\int _{1}^{b}{\frac {1}{at}}(adt)=\int _{1}^{b}{\frac {dt}{t}}=\ln b$ Gandalf61 (talk) 10:24, 3 November 2009 (UTC)

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 people, 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)

What does this number mean?

I was asked to explain the number $\log \left({\frac {i}{e}}+e^{\pi }\right)$ but i cant find a reference to it. What does the number mean? — Preceding unsigned comment added by 70.139.114.18 (talk) 18:47, 25 June 2012 (UTC)

It might be intended as a joke. It has no special significance of which I am aware. You could try asking at the Wikipedia:Reference desk/Mathematics. JRSpriggs (talk) 20:06, 25 June 2012 (UTC)

Error in reference for taylor series

In the reference given for taylor series, http://math2.org/math/expansion/log.htm, I think the formula given for the Taylor Series Centered at 1 is wrong. The funniest is that the formula which is in the Wikipedia article is right, but is giving the link with the wrong formula as reference.

Formula given in the wikipedia article (right):

\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots    

Formula given in the reference (wrong): ln (x) = (x-1) - (1/2)(x-1)2 + (1/3)(x-1)3 + (1/4)(x-1)4 + ... — Preceding unsigned comment added by Asteba (talk • contribs) 18:45, 15 August 2013 (UTC)

Simpler explanation for non-math nerds

I'm trying to figure out what e is, and e is defined using natural logarithm, while natural logarithm is defined using e. It's a self referential definition. Can we put something in here or on the 'e' page that explains what 'e' is or what 'natural logarithm is without using either of those two terms? There must be a simpler way of explaining it, besides giving us the value estimation.Jasonnewyork (talk) 15:01, 30 October 2013 (UTC)

e is the unique real number such that

$\int _{1}^{e}{\frac {1}{x}}\,dx=1\,.$

Is that clear enough? JRSpriggs (talk) 19:11, 30 October 2013 (UTC)

Chemistry Notation

The notation "ln" is not used that much by mathematicians, "log" is much more common. So I find the use here rather odd -- especially since the explanation for the notation is an extract from a chemistry book. — Preceding unsigned comment added by 84.93.54.119 (talk) 11:55, 24 December 2013 (UTC)

I, a mathematician, use "ln" much more than "log". JRSpriggs (talk) 02:02, 25 December 2013 (UTC)

High precision approximation technique using Newton's method does not work.

High precision approximation technique using Newton's method does not work. Example find the approximation of ln(x) with x=3, m=8: s=x*2^m, s=3*2^8, s=768, ln(x)~pi()/(2*average(1,4/s))-8*ln(2) = -2.41986 != ln(3) = 1.0986. — Preceding unsigned comment added by 69.14.214.74 (talk) 16:08, 2 January 2014 (UTC)

You used the wrong average. You used the arithmetic mean when you should have used the arithmetic–geometric mean. JRSpriggs (talk) 09:15, 4 January 2014 (UTC)

A comparison of the series for calculating the natural logarithm

Here, I will apply each of the three series described in the article to solving the same problem, to wit, specify in decimal an interval of length not greater than 0.000050 which must contain ln 1.100000 . The upper bounds are obtained by multiplying the last term (or two terms) calculated by a factor which over-estimates the tail of the series.

1. Series which converges for $\vert x-1\vert \,$ less than 1, i.e. in a disc of radius 1 around 1:

$\ln {\frac {11}{10}}\,=\,{\frac {1}{1\cdot 10}}+{\frac {-1}{2\cdot 10^{2}}}+{\frac {1}{3\cdot 10^{3}}}+{\frac {-1}{4\cdot 10^{4}}}+\ldots \,.$

The lower bound is

$\ln {\frac {11}{10}}\,>\,{\frac {1}{1\cdot 10}}+{\frac {-1}{2\cdot 10^{2}}}+{\frac {1}{3\cdot 10^{3}}}+{\frac {-1}{4\cdot 10^{4}}}\,$

> 0.100000 + -0.005000 + 0.000333 + -0.000025 = 0.095308 .

The upper bound is

$\ln {\frac {11}{10}}\,<\,{\frac {1}{1\cdot 10}}+{\frac {-1}{2\cdot 10^{2}}}+\left({\frac {1}{3\cdot 10^{3}}}+{\frac {-1}{4\cdot 10^{4}}}\right)\cdot {\frac {100}{99}}\,$

< 0.100000 + -0.005000 + ( 0.000334 + -0.000025 ) · 1.010102 < 0.095000 + 0.000313 = 0.095313 .

2. Series which converges for $\left\vert {\frac {x-1}{x}}\right\vert \,$ less than 1, i.e. when the real part of x is greater than 0.5:

$\ln {\frac {11}{10}}\,=\,{\frac {1}{1\cdot 11}}+{\frac {1}{2\cdot 11^{2}}}+{\frac {1}{3\cdot 11^{3}}}+{\frac {1}{4\cdot 11^{4}}}+\ldots \,.$

The lower bound is

$\ln {\frac {11}{10}}\,>\,{\frac {1}{1\cdot 11}}+{\frac {1}{2\cdot 11^{2}}}+{\frac {1}{3\cdot 11^{3}}}\,$

> 0.090909 + 0.004132 + 0.000250 = 0.095291 .

The upper bound is

$\ln {\frac {11}{10}}\,<\,{\frac {1}{1\cdot 11}}+{\frac {1}{2\cdot 11^{2}}}+{\frac {1}{3\cdot 11^{3}}}\cdot {\frac {11}{10}}\,$

< 0.090910 + 0.004133 + 0.000251 · 1.100000 < 0.095320 .

3. Series which converges for $\left\vert {\frac {x-1}{x+1}}\right\vert \,$ less than 1, i.e. when the real part of x is greater than 0.0:

$\ln {\frac {11}{10}}\,=\,{\frac {2}{21}}\left(1+{\frac {1}{3\cdot 441}}+{\frac {1}{5\cdot 441^{2}}}+{\frac {1}{7\cdot 441^{3}}}+\ldots \right)\,.$

The lower bound is

$\ln {\frac {11}{10}}\,>\,{\frac {2}{21}}\left(1+{\frac {1}{3\cdot 441}}+{\frac {1}{5\cdot 441^{2}}}\right)\,$

> 0.095238 · ( 1.000000 + 0.000755 + 0.000001 ) > 0.095309 .

The upper bound is

$\ln {\frac {11}{10}}\,<\,{\frac {2}{21}}\left(1+{\frac {1}{3\cdot 441}}+{\frac {1}{5\cdot 441^{2}}}\cdot {\frac {441}{440}}\right)\,$

< 0.095239 · ( 1.000000 + 0.000756 + 0.000002 ) < 0.095312 .

So these series agree that ln 1.1000 is 0.0953 to four decimal places. Of these three series, the last is the most efficient, that is, it gives the most precision for the same effort, or it achieves the same precision with least effort.

I did not include the so-called high precision methods and continued fractions in this comparison because it is not clear to me how to get error bounds on those methods. If you have a suggestion of how to do that I would be pleased to read it. JRSpriggs (talk) 19:44, 8 January 2014 (UTC)

Here, I apply Newton's method with f(y) = exp(y)-x to the same problem. In general, the iteration is

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

For 0 ≤ y ≤ 0.1, the exponential function can be calculated to six decimal places (actually maybe ten or more places) by the approximation

$\exp(y)\,\approx \,1+{\frac {y}{1}}\left(1+{\frac {y}{2}}\left(1+{\frac {y}{3}}\left(1+{\frac {y}{4}}\left(1+{\frac {y}{5}}\cdot {\frac {1}{1-{\frac {y}{6}}}}\right)\right)\right)\right)\,.$

Also notice that when calculating error bounds on y_n+1, we can neglect any small errors in y_n. To get the error bounds, I will use the facts that

${\frac {x-1}{x}}\leq \ln(x)\leq x-1\,,$

$\ln(x)=y_{n}+\ln \left({\frac {x}{\exp(y_{n})}}\right)\,$

and consequently

$y_{n}+{\frac {{\frac {x}{\exp(y_{n})}}-1}{\frac {x}{\exp(y_{n})}}}\leq \ln(x)\leq y_{n}+{\frac {x}{\exp(y_{n})}}-1=y_{n+1}\,.$

Now, in the case of our problem where x=1.1, we get

$y_{0}=0\,$

$y_{1}=0+{\frac {1.1}{\exp(0)}}-1=1.1-1=0.1\,$

$y_{2}=0.1+{\frac {1.1}{\exp(0.1)}}-1\approx 0.0953\,$

And thus the upper bound is

$\ln(1.1)<y_{3}=0.095300+{\frac {1.1}{\exp(0.095300)}}-1<0.095311\,.$

The lower bound is

$\ln(1.1)>0.095300+{\frac {{\frac {1.1}{\exp(0.095300)}}-1}{\frac {1.1}{\exp(0.095300)}}}>0.095310\,.$

So this might be faster than the series (if x is near 1.0; we want much more precision and you do not put an excessive effort into calculating the earlier values of y), but at the expense of being considerably more complicated. JRSpriggs (talk) 13:15, 14 January 2014 (UTC)

On second thought, since the above version of Newton's method merely has quadratic convergence, it might be better to use a different function with Newton's method which will have cubic convergence, to wit

$f(y)=x\exp(-{\frac {1}{2}}y)-\exp({\frac {1}{2}}y)\,.$

This would give

$y_{n+1}=y_{n}-{\frac {x\exp(-{\frac {1}{2}}y_{n})-\exp({\frac {1}{2}}y_{n})}{-{\frac {1}{2}}x\exp(-{\frac {1}{2}}y_{n})-{\frac {1}{2}}\exp({\frac {1}{2}}y_{n})}}\,$

$=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}\,.$

If we replace

$y_{n}=\delta _{n}+\ln(x)\,,$

then we get

$\delta _{n+1}=\delta _{n}+2\cdot {\frac {1-\exp(\delta _{n})}{1+\exp(\delta _{n})}}\,.$

We recognize this as the hyperbolic tangent

$\delta _{n+1}=\delta _{n}-2\cdot \tanh({\frac {1}{2}}\delta _{n})=+{\frac {{\delta _{n}}^{3}}{12}}-{\frac {{\delta _{n}}^{5}}{120}}+{\frac {17{\delta _{n}}^{7}}{20160}}-\cdots \,.$

This has cubic convergence as one might expect since the second derivative of function f is zero at the same y as f is zero:

$f''(y)={\frac {1}{4}}f(y)\,.$

Again for x=1.1, we get

$y_{0}=0\,$

$\delta _{0}\approx {\frac {-1}{10}}\,$

$y_{1}=0+2\cdot {\frac {1.1-1}{1.1+1}}={\frac {2}{21}}=0.095238\;095238\;095238\ldots \,$

$\delta _{1}\approx {\frac {-1}{12\;000}}\,$

and at the next step we would have

$\delta _{2}\approx {\frac {-1}{20\;736\;000\;000\;000}}\,$

which would give us more precision at the second step than the previous function f gave us at the third step.

Thus, barring errors in my calculation of the exponential, this should be ln(1.1) correct to 13 decimal places:

$y_{2}=0.095\;310\;179\;804\;293\;4\ldots \,.$

JRSpriggs (talk) 10:35, 29 January 2014 (UTC)

According to my copy of the Handbook of Mathematical Functions, the actual value is

$\ln(1.1)=0.095\;310\;179\;804\;324\;9\ldots \,.$

Thus the actual error is -315×10⁻¹⁶ which is slightly closer to zero than my estimate of the error. JRSpriggs (talk) 11:18, 29 January 2014 (UTC)

Natural logarithm of thirty

Calculating the natural logarithm of a number distant from 1, e.g. 30, is a problem for the methods described in my previous section of talk. The Taylor series does not converge there at all. The other two series for calculating the natural logarithm of 30 and the series for the exponential of ln(30) would converge, but only extremely slowly. The continued fractions require as many steps as the corresponding series from which they were derived. So as a practical matter, it is impossible to directly calculate the natural logarithm of thirty by any of those methods. However, it can be done indirectly by factoring 30 appropriately. For example:

$30={\left({\frac {81}{80}}\right)}^{15}\cdot {\left({\frac {16}{15}}\right)}^{34}\cdot {\left({\frac {25}{24}}\right)}^{25}\,$

which is easily verified by counting up the factors of 2, 3, and 5 (these are the only prime factors of these numbers). Thus

$\ln(30)=15\cdot \ln {\left({\frac {81}{80}}\right)}+34\cdot \ln {\left({\frac {16}{15}}\right)}+25\cdot \ln {\left({\frac {25}{24}}\right)}\,.$

These numbers are close enough to 1 that their series converge quite quickly. A low precision value can be obtained with just the first term of the third series, so:

$\ln(30)\approx 15\cdot {\frac {2}{161}}+34\cdot {\frac {2}{31}}+25\cdot {\frac {2}{49}}=0.1863+2.1935+1.0204=3.4003\,.$

Error analysis gives 3.40 < ln(30) < 3.41 . JRSpriggs (talk) 17:37, 6 February 2014 (UTC)

JR, this is good stuff, but I don't see what it has to do with improving the article. I don't think there's any prospect that we're going to put detailed info about the challenges for numerical methods, and ways to get around them, into the article itself. --Trovatore (talk) 19:55, 6 February 2014 (UTC)

Yes, I just could not resist the challenge of seeing how far I could go using just the simple calculator built into Windows. It can remember only one other number than the one that I am currently working on and has no functions more complex than square-root.

Also, I still have doubts about the usefulness of the so-called high precision methods and the continued fractions.

I did make some changes to the article between January 17 and February 3 based on things I learned while writing the previous section of talk. JRSpriggs (talk) 20:53, 6 February 2014 (UTC)

Shouldn't the axes of the graph, "Graph of the natural logarithm function" be labelled?

I feel the graph, "Graph of the natural logarithm function", would be improved by labeling the axes. — Preceding unsigned comment added by 71.166.8.154 (talk) 03:43, 26 February 2014 (UTC)

One plot at bottom of page is wrong.

The plot labelled |Im ln(x+iy)| at the very bottom of the page cannot possible be right: it fails to show the branch cut! That is, there should be a discontinuity of 2\pi i across the cut, so that visually, it looks like one turn of a helix or screw (e.g. the screws they use on construction sites to dig boreholes). the correct screw-plot is shown on complex logarithm 67.198.37.16 (talk) 05:17, 1 December 2014 (UTC)

The absolute value would have the effect of equating the + \pi i and - \pi i, would it not? JRSpriggs (talk) 09:18, 1 December 2014 (UTC)

definition

A definition that does not use e is helpful, since e is itself defined in terms of ln. But I don't quite get the part about "The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a." The natural logarithm is an area? --Richardson mcphillips (talk) 21:42, 17 March 2015 (UTC)

Yes, the "area under a curve" is the definite integral of a function, in this case 1/x. It seems to be adequately described and linked in the article. —Quondum 21:53, 17 March 2015 (UTC)

Logarithmus naturalis

I undid this change for two reasons.

One is the unexplained use of "aka". I think that's too informal for encyclopedic use. Even if it weren't, it's my understanding that it's not widely understood outside the United States, so we should prefer to avoid it per WP:COMMONALITY.

That point, though, could easily have been fixed. The question is, do we want to call out "logarithmus naturalis" in the first sentence at all?

I am skeptical, but am open to be convinced. Can anyone point me to modern uses of this term in the wild? --Trovatore (talk) 21:22, 5 October 2015 (UTC)

I am also skeptical. Good revert, as far as I'm concerned. -Starke Hathaway (talk) 21:35, 5 October 2015 (UTC)

Per WP:ONLYREVERT, reverting a constructive contribution is never a good thing unless it makes the article worse or is even harmful, because the information is wrong etc. Otherwise, improving on the previous edit(s) is the proper way to go. --Matthiaspaul (talk) 22:28, 5 October 2015 (UTC)

ONLYREVERT is an essay. It's one that I think is quite completely incorrect, basically just about as wrong as it could possibly be. See instead WP:BRD, which is also an essay, but a much better one. --Trovatore (talk) 22:31, 5 October 2015 (UTC)

If practised with the necessary care among experienced editors, BRD can be a mechanism to allow fast progress and ensure quality. The problem with the BRD essay is that quite a few (even experienced) editors incorrectly assume BRD would allow them to revert anything on sight they do not like for what reasons ever instead of even trying to improve on it (even though WP:FIXTHEPROBLEM, which is policy, states otherwise), and quite a few of these editors do this habitually - this is a great waste of resources and detrimal to progress, and the climate it creates is IMO one of the reasons why the English WP is losing so many good editors.

While I can accept BRD, there would have been more appropriate alternatives, f.e. improving the "aka" to what you consider a more encyclopedic tone, leaving an edit summary stating your concerns regarding the term logarithmus naturalis, or opening a discussion here without reverting me first.

--Matthiaspaul (talk) 03:12, 6 October 2015 (UTC)

You don't understand. The "aka" was a secondary issue, easily fixable. What I was skeptical about was mentioning the Latin phrase in the first sentence at all. It needed to be discussed.

Discussions should normally take place using the status quo ante. The ONLYREVERT essay is completely wrong about the bias being to accept the edit. The burden of proof is on the change (except in certain cases, such as the removal of uncited controversial claims — in that case, the burden is on the side that wants the claim to appear). --Trovatore (talk) 04:02, 6 October 2015 (UTC)

There is no burden of proof here. The fact that the natural logarithm is also called logarithmus naturalis is hardly disputable. --Matthiaspaul (talk) 15:32, 18 November 2015 (UTC)

logarithmus naturalis is the de-facto international name of the function. It is understood among mathematicians worldwide, whereas natural logarithm is just a translation into a specific language. The idea to chose Latin names for mathematical functions was to be independent of any particular target language, because mathematical concepts are considered universal and not locale-specific as well. Therefore I think it is important to mention it in the lede, not just somewhere in the history section as if it was something archaic. Also, it helps to motivate the symbolic function name ln(), which was derived from logarithmus naturalis as well.

Regarding "aka", I used it seeking for a short form and because it is also used in uncountable of other articles. I'm open for suggestions how to improve on it, of course. "Natural logarithm (or logarithmus naturalis) ..." would be fine as well.

--Matthiaspaul (talk)

This is the English Wikipedia. What is relevant is the extent to which logarithmus naturalis is used in English. My estimation is that that extent is not sufficient to merit a mention in the first sentence. Can you demonstrate otherwise? --Trovatore (talk) 22:34, 5 October 2015 (UTC)

Actually, the English Wikipedia is an international project, not restricted to North-American or British views only. What is relevant in an article about a mathematical function is mathematics and related information - like the name. The language is just a vehicle to transport the message.

I cannot provide statistics, but Google turns up with enough hits using "logarithmus naturalis" also in very recent English literature. It's certainly not as common as "natural logarithm" in the English literature today, but that's just normal and hardly a reason not to mention this alternative name as we usually do in the lede (MOS:LEAD) of articles per WP:LEADSENTENCE, MOS:BOLDSYN and MOS:FORLANG.

--Matthiaspaul (talk) 03:12, 6 October 2015 (UTC)

I do not believe the Latin phrase has enough currency in the English-language mathematical literature to warrant a mention in the first sentence. It's not about North America or Britain, but it is about English. We would not give the French or German or Italian names in the first sentence, and I see no reason to give the Latin one either; it might make sense if Latin were still the lingua franca of science and mathematics, but it isn't. --Trovatore (talk) 04:05, 6 October 2015 (UTC)

(By the way, I looked up MOS:FORLANG, and it says [i]f the subject of the article is closely associated with a non-English language.... Is the natural logarithm closely associated with Latin? That's easily answered. No, it is not closely associated with Latin.) --Trovatore (talk) 04:09, 6 October 2015 (UTC)

To the contrary, I do think it is quite closely associated with Latin by the very fact that Latin was the universal language of science for centuries and many mathematical concepts and functions therefore carry Latin names. While we no longer express ourselves in Latin directly in sciences, the Latin names are still quite standard and are commonly used and understood - at least by the older ones who were regularly taught Latin in school. Not mentioning the Latin name in the lede is a form of recentism and bias (which doesn't look good at all in an encyclopedia). --Matthiaspaul (talk) 15:32, 18 November 2015 (UTC)

I'm sorry, you're just wrong. The Latin name is not commonly used in mathematics in English. --Trovatore (talk) 20:12, 18 November 2015 (UTC)

Log as time

At the mathematical website BetterExplained, Kalid Azad has a financial interpretation of natural logarithm: he writes, "Natural logarithm gives you the time needed to reach a certain level of growth." This article is about the mathematical function, apart from any temporal or financial situation. So the following external link was removed:

Demystifying the Natural Logarithm (ln) | BetterExplained

Since the explanation is quite useful in context, it was used for a reference in the article Rule of 72 which Azad explains in his article. — Rgdboer (talk) 01:28, 8 December 2015 (UTC)

Some more proofs

I noticed that the proofs section is kind of thin. Could we have proofs of some of the other properties? In particular, why is it that the limit as "n" approaches 0 of (x^n-1)/n is ln(x)? I've tried to find the proof but I don't know where to look.98.197.193.213 (talk) 15:35, 4 June 2017 (UTC)

Why natural ?

Unit area when b = e as exploited by Euler.

The section "Origin of the term" discusses the curve (x, log x) and notes that the slope is exactly one only for the natural logarithm. The unreferenced claim is made that

While the methods for computing the value of e are fascinating from various mathematical perspectives, they all can be thought of as resulting from the pursuit of this condition.

Instead of the slope property, the origin of the naturalness of this logarithm derives from the unit area found under the "natural hyperbola" y = 1/x between x = 1 and x = e. The unit area property also holds for the hyperbolic sector determined by these x values. Area of a hyperbolic sector defines the corresponding hyperbolic angle, which the precursor to circular angle. As the area measure gives substance to the natural logarithm, and the illustration associates e with unit area in a natural way, it is proposed that this section be re-written on the basis of area under the hyperbola. — Rgdboer (talk) 03:26, 11 November 2017 (UTC)

I think you have the clear burden of proof here. There are two things you must demonstrate before it makes sense to change the article along these lines:

First, that the term "natural hyperbola" is standardly used in this way, and

Second, that this is the reason for the "natural" in "natural logarithm"

To be honest, I don't think you're going to be able to meet even the first burden. A Google search for "natural hyperbola" in double quotes has only two relevant hits on the first page, and they both trace back to you, at least assuming you use the same name on Wikibooks (this page appears to be written entirely by you). But if you have sources to share that prove your assertions, by all means please do so. --Trovatore (talk) 05:13, 11 November 2017 (UTC)

The power series for calculating logarithms all depend on the base being e. So the natural thing is to define the natural logarithm first and then define the other logarithms in terms of it. JRSpriggs (talk) 06:27, 11 November 2017 (UTC)

Leonard Euler's Introduction to the Analysis of the Infinite (1748) refers in chapter 7, section 122, to the natural or hyperbolic logarithm, "since the quadrature of the hyperbola can be expressed through these logarithms." Euler is referencing the work of Gregoire de Saint Vincent in 1647 on quadrature of the hyperbola. According to V. Frederick Rickey, Euler used the phrase "denotes that number whose hyperbolic logarithm is 1" for e, and in particular in a letter to Goldbach November 25, 1731.

Euler’s chapter 7 introduces his use of e. His approach to the natural logarithm is prepared in chapter 4 introducing infinite series, chapter 5 mentioning transcendental functions, and chapter 6 on power functions a^z. The inverse of a power function is a logarithm. Chapter 7 derives the infinite series for power functions, and gives the expression for the exponential function inverse to natural logarithm. Trigonometry only arises in chapter 8: "Transcendental quantities that arise from the circle."

Thus the argument does not stem from a natural hyperbola xy = 1, but from G. de St. Vincent and Euler in the 17th and 18th centuries. The visual representation of natural logarithm as an area is founded in history, and provides novices with a concrete image. — Rgdboer (talk) 03:03, 12 November 2017 (UTC)

I would both agree with Rgdboer that the explanation cited is not particularly insightful; and would also note that he seems to. conspicuously, be the one here that is providing evidence-based arguments concerning both the actual historical basis and philosophical roots of the natural log. The article certainly could use some help in that regard. Wikibearwithme (talk) 07:52, 13 January 2018 (UTC)

As was pointed out at the outset of in Euler's original paper introducing the number e, which "therefore denotes the base of the natural, or hyperbolic, logorithm (translated)." This seminal introduction appears to quite explicity indicate the hyperbolic origin of the term "natural." Wikibearwithme (talk) 06:10, 14 January 2018 (UTC)

Other articles such as Nicholas Mercator, History of logarithms, History of calculus, James Gregory (mathematician) bear on this subject. Rapid developments at the end of the 17th century brought about this innovation in mathematics that made calculus well-rounded with an expression for quadrature of the hyperbola. Composing an alternative paragraph or section to that now in the article requires some sensitivity to the extremely high-profile of this article (top 500) and the expressed interest of other editors. Thus dialogue here in Talk is welcomed to avoid or minimize reverts in the article when change is made. WP:BOLD needs to be tempered for high-profile sites where there are dug-in viewpoints with defenders. — Rgdboer (talk) 19:39, 14 January 2018 (UTC)