Talk:Logarithm/Archive 4

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CORDIC?

Someone recently added a section about computing logs using a "CORDIC-like" method. I don't doubt that one can compute it this way, but I'm not sure whether this is notable enough (i.e., efficient enough) to be mentioned, especially at this length. Unless we find a few references backing this up, I'm inclined trim down the section to one sentence at most (or even remove it). Jakob.scholbach (talk) 21:58, 27 July 2011 (UTC)

I probably believe it should be described, but isn't there a CORDIC page for it? Then a link to that page, along with a small description, would seem right to me. Gah4 (talk) 00:53, 27 September 2011 (UTC)

As far as I know the CORDIC method is the method for computing the logarithm in hardware, because it really only needs shifts, adds, a small lookup table (n constants for n bits precision), no multiplication, no division, no square root or anything more complicated. You get one bit precision per iteration. You can apply the same tricks like coefficient estimations for faster addition that are used in Pentium's FDIV. That is, logarithm computed this way is as fast as division, because it is essentially a division algorithm. Btw., the same is true for exponentiation, square root, trigonometric functions and their inverse functions. HenningThielemann (talk) 07:49, 28 July 2011 (UTC)

If that is true it would not be hard to provide sources to support that.TR 08:18, 28 July 2011 (UTC)

I have taken the CORDIC algorithm from the German book "Arithmetische Algorithmen der Mikrorechentechnik" by Jorke, Lampe, Wengel, 1989. They claim that Briggs used the method for calculating logarithm tables and that the method is used in pocket calculators. To this end they refer to a reference numbered 6.20, but the reference list ends at 6.19. :-( Then they use the method for implementing logarithm, exponential and trigonometric functions on a Z80, an 8-bit processor without hardware multiplication. There is a long list of references about CORDIC, several of them describing hardware implementations. On the other hand, I cannot find this method in glibc. I got the source with "apt-get source libc6" and searched for logarithm. Of course, they use FPU logarithm, if available. If not, they use rational function approximations (ieee754/ldbl-128/e_log2l.c) or Newton method on the exponential function (ieee754/dbl-64/mplog.c). Then there is New algorithms for improved transcendental functions on IA-64, where the authors describe the implementation of FYL2X in Intel processors. They use argument reduction by a lookup-table and perform Taylor approximation on the remainder. That is, their algorithm needs some multiplications, but it does not seem to hurt performance or space on the chip. HenningThielemann (talk) 20:34, 28 July 2011 (UTC)

You shouldn't look for software or desktop CPU implementations. If CORDIC algorithms are used somewhere then only where size/cost of implementation is important (and speed less), e.g., FPGA cores. Nageh (talk) 21:20, 28 July 2011 (UTC)

As far as I know, CORDIC, or the decimal for of it, is commonly used in scientific hand calculators. (or at least it was in the days when memory was more expensive than it is now.) Many references go to the HP scientific calculator series. Gah4 (talk) 00:53, 27 September 2011 (UTC)

I'm indifferent to the CORDIC section. The article is about logarithms, but the section is more about computer implementation. It belongs somewhere; maybe here; maybe somewhere else. I am troubled by the apparent WP:COPYLINK violation. There's no indication that jacques-laporte.org has a right to republish an IBM Journal article. Glrx (talk) 20:56, 28 July 2011 (UTC)

I trimmed down that section. The given reference is old, apparently primary source (as opposed to secondary source, which is preferrable), and does not establish the degree of notability needed to devote this much space. Also, this more recent and more exhaustive book, which does talk a lot about CORDIC and logarithms, does not mention the two together. Jakob.scholbach (talk) 07:03, 29 July 2011 (UTC)

Ahem - please allow me to correct some things:

• "given reference is old" - absolute age does not matter when it comes to math and algorithms.
• "apparently primary source" - please reread WP:PRIMARY. The author (primary source) wrote a rather scholarly research paper (even citing sources) published in the IBM Journal (the secondary source). Given that the IBM Journal had an editorial staff and policies, that makes it a reliable source, too.
• "degree of notability" - a careful rereading of WP:N will show that "notability" applies to whole articles, absolutely not claims within articles. For claims, verifiability and reliability apply, not notability per se.
• It matters little that one source does not mention CORDIC and logarithms together

A good reason for trimming the section is relevance: that it strays from CORDIC into "CORDIC-like" territory. For that reason, I agree with the trimming.

By the way, the book you link to cites the IBM Journal article - search in it for "pseudo".

--Lexein (talk) 16:12, 29 July 2011 (UTC)

History

The Google's Computer history Museum mentions ([1] and [2]) quite a few new names in connection with the history of invention of the logarithm. They also have several pictures they appear to be willing to share provided they are credited. // stpasha » 06:41, 8 September 2011 (UTC)

Napier is already mentioned here as the main inventor of logarithms and Oughtred is mentioned as the inventor of the slide rule. There is a picture of Napier in this article and a picture of Oughtred in his article and another one at the slide rule article. Dmcq (talk) 11:58, 8 September 2011 (UTC)

I think this page needs cleaning up to be consistently y=f(x) rather than x=f(x)

There's inconsistency on this page between discussions of 'x=log(base b)y' versus 'y=log(base b)x', which I think causes confusion (it certainly does to me).

I think it would be great if someone (more experienced in wikipedia editing than me (including awareness of all the rules etc) plus in-the-loop on this page and its writing) could tidy it so that one can read the page without having to mentally reverse the x and y in all parts of the discussion (including the definition) which talk about 'x=log_{(base b)}y'. I don't see why on a page about logarithms it should be put that way, when that way doesn't align to people's usual graphical conceptions (y=f(x), where in this case the logarithm is what's being explored)) etc. And, as I say, the inconsistency makes things even harder to folk like me.

Tfll (talk) 17:33, 23 September 2011 (UTC)

Could you be more specific about which statement in the article you think is inconsistent with elsewhere thanks. Dmcq (talk) 21:15, 23 September 2011 (UTC)

---

Jim Bandlow (talk) 04:09, 28 September 2011 (UTC)
If one starts off with an exponential function Y = b^X, then solves for X, you get Log_bY = X,
with Y appearing as the independent variable, which is confusing.
It's confusing since it's actually a transposed exponential function, and both are the "mirror" inverse of the Log function Log_bX = Y.
Antilog form is X = b^Y, just another form of the Log function, with both forms having the same plot.
An antilog is not a real inverse; it's just the Log function solved for X instead of Y.

Thanks I can see the problem now in the definition where they do that. It does go against how things are done normally so I'll fix it. Dmcq (talk) 07:59, 28 September 2011 (UTC)

article logarithms

This is indeed a very good article, laymen are able to follow it step by step, no advanced knowledge required. This is contrary to many other articles concerning mathematics which are clearly written for the mathematical community and belong to proceedings of a math departement of a university rather than wikipedia

regards,

Tjerk Visser Amsterdam The Netherlands — Preceding unsigned comment added by 80.57.154.203 (talk) 08:31, 29 September 2011 (UTC)

Complex logarithms

There are some issues in the section "Calculation" related to complex numbers. Specifically, the logarithm of complex numbers is allowed for in the discussions about Gelfond–Schneider theorem and Area hyperbolic tangent. However, the complex logarithm is discussed only later in the section "Generalizations". Moreover, in the subsection "Taylor series", the variable is denoted by z although it is assumed to be real. If this is the case, it is more natural to denote it by x (and by z in the footnote). Isheden (talk) 17:06, 5 January 2012 (UTC)

That bit isn't calculation anyway so it should be somewhere else or in its own section.

also this article is not about complex logs, there's another article about them. There is no real need to bring in complex numbers at that point anyway. The main thing is that things like log of 3 to base 2 are transcendental and bringing in complex logs doesn't add all that much at this level, the article about the Gelfond-Schneider theorem article can go into it in more detail. Dmcq (talk) 17:18, 5 January 2012 (UTC)

I agree. The theoretical discussion should be moved out of the section "Calculation". The series should be stated in terms of real numbers in this article. The complex versions can be stated in the complex logarithm article. Isheden (talk) 21:29, 5 January 2012 (UTC)

After thinking further about this and checking the reference provided, I believe it would be more natural to present the original statement of the theorem (see Gelfond–Schneider theorem) in the Exponentiation article (section "Powers of complex numbers") instead of here. What the reference says is: "The Gelfond–Schneider theorem shows that for any non-zero algebraic numbers $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, $\beta _{2}$ with $\log \alpha _{1}$, $\log \alpha _{2}$ linearly independent over the rationals, we have $\beta _{1}\log \alpha _{1}+\beta _{2}\log \alpha _{2}\neq 0$." I'm not sure what this has to do with calculation of logarithms. Are there any objections to my proposal? Isheden (talk) 22:52, 7 January 2012 (UTC)

I think that the theoretical result that the natural logarithm of a rational number other than 1 is always transcendental is IMO directly relevant to this article. I think that there may be a more direct theorem stating this. Other than what is needed for this result, the mention of the Gelfond-Scheider theorem could/should be removed. And it definitely does not belong under Calculation. — Quondum^t_c 05:02, 8 January 2012 (UTC)

Good point. According to Transcendental number#Numbers proven to be transcendental, this result follows from the Lindemann–Weierstrass theorem. Perhaps it should be mentioned in the section "Analytic properties" that the logarithmic function is transcendental together with the result on the transcendental values? Isheden (talk) 09:51, 8 January 2012 (UTC)

I'm out of my depth at his point. I do not even know whether the transcendental nature of the function is called an analytic property. — Quondum^☏✎ 14:31, 8 January 2012 (UTC)

I have moved the discussion about the Gelfond–Schneider theorem to a more appropriate place. Can someone clarify if the Gelfond–Schneider theorem is needed for logarithms to other bases than e? Isheden (talk) 14:14, 8 January 2012 (UTC)

At this point, where basic analytic properties are explained, the discussion could be highly confusing to the layman. I propose that it be moved to a subsection of its own at the end of section Analytic properties, possibly named Transcendence of the logarithm. Nageh (talk) 18:22, 8 January 2012 (UTC)

Done. Isheden (talk) 18:44, 8 January 2012 (UTC)

Suggested addition

Something that might be interesting to see is the following. Suppose a source is presenting data in the form:

$\log b=c\pm \Delta c$

where b is some parameter such as stellar luminosity or gravity. The article could show how one can derive an approximation f(c, Δc) to the equivalent error range Δd that satisfies:

$b=10^{c}\pm \Delta d$

$\Delta d\approx f(c,\Delta c){\text{ for sufficiently small }}{\frac {\Delta c}{c}}$

This could perhaps go in the derivatives section. I'm not sure where you'd source this from though.

An example of where this might occur is in astronomy, where the logarithm of the luminosity of a star relative to the sun is listed as a value plus an error range, and you wanted to convert that into an actual luminosity plus or minus an error range. (See also Parallax#Parallax error.) Regards, RJH (talk) 00:07, 7 January 2012 (UTC)

The transfer function of a small error range is useful for any smooth function, and is not specific in any way to logarithms. It would thus not be appropriate in this article, but may have a place in, for example, Derivative. — Quondum^t_c 06:28, 7 January 2012 (UTC)

Yes, a general solution could be covered on the derivative article. But the solution for the logarithm would be unique, so it is appropriate to cover that here. Otherwise, why cover Taylor series here since the same logic applies? Regards, RJH (talk) 06:37, 7 January 2012 (UTC)

You may have a point. Personally I favour minimal repetition between articles but I may stand alone on this. The form of the transfer function for the error in the case of the logarithm (via the derivative) is particularly simple. — Quondum^t_c 07:19, 7 January 2012 (UTC)

Let's start with a source that covers this and see where it takes us. Dicklyon (talk) 07:51, 7 January 2012 (UTC)

Okay I tracked it down. It's actually already covered in the "Propagation of uncertainty" article. Thanks. Regards, RJH (talk) 06:17, 17 January 2012 (UTC)

Knowledge of logarithms helps in understanding the relative magnitude of things. If it's twice as much add 0.301 to the logarithm, If you want parsecs, reduce the log of the light year value by 0.51. A year is 86400 x 365.25 = 10E^7.499 seconds say 10E^7.5 and light velocity is 299,792,458 meters/second or10E^10.4768 centimeters/second so a lightyear is 10E^17.975 or say 10E^18 cm.WFPM (talk) 03:31, 16 March 2012 (UTC)

Cannot Verify all of the the methods described here.

I am changing this posting by following the instruction on how to post a new comment. I am new to this and posted my comments incorrectly at first.

My comment about this page is that... I have been trying to verify the functions on this page by building computer functions that implement the formulas given. I have been able to replicate several, but not all the formulas. For example, I have been able to replicate the Taylor method, and the one described with "More efficient series", within the limits mentioned in the article. The Arithmetic-Geometric mean calculation, however, is one that I have not been able to replicate. There is something missing in the presentation. Not sure what it is. I went back to the article that is referenced, and it too is unclear. The referenced source describes an algorithm that appears to me to be different from the one presented here. Statguy1 (talk) 02:49, 17 March 2012 (UTC)

I agree, the formula is wrong, doesn't work, and isn't in agreement with the linked source. Interpreting the source to fix it will need more work. I goofed. It's actually does work, when I fix my matlab code to get it right. Email me if you want me to send code that works to verify it. Dicklyon (talk) 06:03, 17 March 2012 (UTC)

That is good news. Thank you for checking. I don't have Matlab, so my work is in VB and C and Javascript. There is always the challenge of making the translation to one of those tools. Is there a way I can send you an email privately? Statguy1 (talk) 15:35, 17 March 2012 (UTC)

Yes, if you're on my user page or talk page your toolbox should have a "email this user" link. Dicklyon (talk) 18:33, 17 March 2012 (UTC)

Thank you for your assistance, today. I am now convinced that the formula under the heading "Arithmetic-geomatric mean approximation" is an accurate description of the method and that it does work properly. Statguy1 (talk) 03:34, 18 March 2012 (UTC)

ln(x) notation

This article says that the notation ln(x) was invented by Irving Stringham, and cites Stringham's 1893 book "Uniplanar Algebra." However, I have found an earlier reference that may be relevant. In Anton Steinhauser's "Lehrbuch der Mathematik," published in 1875, on p. 277, Steinhauser suggests that the natural logarithm of a number a should be denoted "log. nat. a (spoken: Logarithmus naturalis a) or ln. a" Steinhauser then seems to stick to the log. nat. notation, but perhaps Stringham got the idea for using "ln" from Steinhauser. If so, this would explain the motivation for the notation: "ln" is short for "log. nat." which is short for "logarithmus naturalis." Steinhauser's book can be found here: http://books.google.com/books?id=ZzU7AQAAIAAJ

Dan Velleman 174.63.17.86 (talk) 16:32, 29 March 2012 (UTC)

That's interesting. Do you have a copy of the book? Maybe you could scan a few pages around the relevant page and email me? Jakob.scholbach (talk) 14:59, 20 July 2012 (UTC)

No need for that. Click on the books.google.com link he gave and then on the book image. The entire contents are on line. The section he refers to is on page 277, paragraph 439. Steinhauser lists ln as one of several possible abbreviations.--agr (talk) 15:16, 20 July 2012 (UTC)

I don't know why, but I don't see any preview or anything at the google page. Could you help out? Thanks, Jakob.scholbach (talk) 18:09, 22 July 2012 (UTC)

Do you get a "preview this book" link? or a search box? The books display differently in different countries, depending on copyright laws. What's public domain in the US may not be everywhere. Dicklyon (talk) 21:01, 22 July 2012 (UTC)

I don't see either preview snippets nor a search box. I just see the title and such raw data. I also assume it's because of copyright reasons. But if you can see it in the US, you could extract a "screenshot" of the book's relevant page and we can refer to it in the article. Jakob.scholbach (talk) 08:28, 23 July 2012 (UTC)

See if this link works. If not, email me and I'll respond with a screen grab of the page. I suppose we could upload it as an image to WP, too. Dicklyon (talk) 15:51, 24 July 2012 (UTC)

Jargon in the lead paragraph

Correct me if I'm wrong but isn't incomprehensible jargon a no-no on WP, especially in the lead paragraph. I don't know about you, but I don't know many people who would understand this sentence: 'The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number' What's an exponent? Non-mathmeticians are supposed to be able to access and understand this page, this represents a failure on the part of WP, since someone coming hear searching elucidation will no find it. - 86.42.245.86 (talk) 01:54, 20 July 2012 (UTC)

I found it EXTREMELY accessible. In fact I came to this page to congratulate and thank the author for such a concise and clear explanation. Oh and I'd like to point that I have no mathematical background, other than basic high school math. — Preceding unsigned comment added by 108.214.136.201 (talk) 23:37, 1 August 2012 (UTC)

You are right that jargon should be avoided whenever possible. However, the first sentence of the lead should give a concise definition of the subject whenever possible. Achieving both goals in the first sentence is a common difficulty in articles on mathematics. To avoid jargon, I would suggest changing the first sentence to something like this:

The logarithm of a number is the power to which a given value has to be raised to yield that number. Isheden (talk) 10:21, 20 July 2012 (UTC)

Before changing that, please read the FAC discussion about that very sentence. It took a long time to figure out that consensus. In particular, removing the words exponent and base will break the link to the following sentences. I suggest keeping it as it is. Jakob.scholbach (talk) 11:56, 20 July 2012 (UTC)

Yes, before changing the first sentence at all we should definitely discuss the implications here. Naturally, the terms base and exponent have to be introduced before they are used to in an example. Could you provide a link to the FAC discussion you're referring to? Isheden (talk) 12:47, 20 July 2012 (UTC)

Wikipedia:Featured_article_candidates/Logarithm/archive1 --Jakob.scholbach (talk) 13:16, 20 July 2012 (UTC)

In the FAC discussion, several persons raised concerns that the lead sentence is difficult to follow since it relies on knowing what base and exponent are. I don't think that base is very problematic since the meaning is described before the term is mentioned. The question is whether the term exponent can be left out of the first sentence. Exponent is not used anywhere else in the lead (and in fact only a couple of times in the article), so it is not clear why it is needed in the first sentence. Also, the formal definition of logarithm relies on the term exponent without introducing it first. Is the reader assumed to be already familiar with exponentiation? Isheden (talk) 19:08, 20 July 2012 (UTC)

I would suggest a different approach, something like: "The logarithm is a mathematical operation that inverts the process of exponentation or raising a number to a power. In other words, the logarithm of x with base b is the value, y, such that b raised to the y power equals x." It provides the reader with a simple take-away message, log is the inverse of exp, before getting bogged down in details--agr (talk) 20:54, 20 July 2012 (UTC)

The first sentence is good since it gives a concise definition without introducing jargon. In the second sentence, the necessary terminology should be introduced, but I think it is unnecessary to introduce algebraic symbols. Instead, a slightly modified version of what is presently the first sentence would work well. Isheden (talk) 21:26, 20 July 2012 (UTC)

I disagree. The logarithm is, in the first place, merely a number and not an operation. Such meta-understanding comes much later when one has a firm understanding of the basics, I think.

@ Isheden: I'm not entirely opposed to getting around the word exponent (even though it is a prerequisite to know what exponentiation (hence exponent) is. Also this is explained in some detail in the first section, which is the place to offer a more detailed explanation). However, it is necessary to avoid the word "number" too often (and especially messing up base and exponents). Another common mistake (committed by myself, too, earlier): the base is raised by an exponent, and raised to a power. Jakob.scholbach (talk) 08:34, 23 July 2012 (UTC)

It could well be that viewing the logarithm as a number rather than as an operation demands a higher level of mathematical maturity. Beginners tend to think of "taking the logarithm" of a number as the inverse of raising to a power, although they might not know that "raising to a power" is referred to as exponentiation. I don't think you can assume that such a beginner is comfortable with the terms base and exponent. Isheden (talk) 20:51, 23 July 2012 (UTC)

I agree. Our article on Exponentiation defines it as a mathematical operation. I see no reason not to follow that example here.--agr (talk) 00:59, 24 July 2012 (UTC)

Again, I disagree to introducing vague words as "operation" into the first sentence. Really, the decimal log of 1000 is a number, namely 3. Full stop. A logarithm is not some process. If anything it is the result of a process. However, given the need of brevity and conciseness, we just have to define it in a clear and short way in the first sentence. Please do read the FAC discussion: in my understanding it is not possible to rely on absolutely nothing (esp. not on exponentiation), yet explaining logs properly. Also, note that the info about reversing the exponentiation is covered many times in the article proper and also at the end of the lead section.

That said, I'm not digging my heels in as far as the word "exponent" is concerned. If you find a good wording that stays close to what we have, is short and simple, and eliminates that word, fine with me. However, do leave the word base, we need it 3 times in a row in the following sentences.

Finally, I'd like to point out that many other articles (including exponentiation!, whose lead section is certainly not a good example to point at) are much more worth the effort than this one. All of your edits are good-faith, but I and others have invested lots of time in this one here and I'm afraid many edits are at best partial improvements. (in all due respect). For example, Isheden's last edit removed (by accident? the sentence "The third power of some number b is the product of 3 factors of b." I regard this as a step back: making the transition from 2^3 to b^x in two steps rather than one is better, in my view. Jakob.scholbach (talk) 08:36, 24 July 2012 (UTC)

I strongly second the sentiment that efforts are much better aimed at other articles, instead of this one, which is in pretty good shape and the wording of the lede has been hammered out by a small army of editors in the past. Of course, improvements are possible, but most will be very marginal compared to the effort involved.TR 09:23, 24 July 2012 (UTC)

The logical way to introduce the logarithm is as the inverse operation of exponentiation, however the first sentence should define the subject, so I agree that it should be described there as a number. However, the lead should also avoid jargon, which is why the terms exponent and base should not be used without being introduced. Since the meaning of exponent is not explained until later, in my view the word should be avoided in the first sentence.

Of course there are other articles that have more potential for improvement. For example, I would be interested in advice from experienced mathematics editors how to advance function (mathematics) and fraction (mathematics) to GA and finally FA status. Nevertheless, if someone raises a valid issue regarding a featured article, why should we avoid addressing it just because a lot of effort has already been spent in the past?

Following your critique, I reinserted the sentence. Now the transition from 8^3 to b^y has the intermediate steps b^3 and b^n, although the transition from b^n to b^y is the more interesting one in my view. Isheden (talk) 11:09, 24 July 2012 (UTC)

Bringing function (mathematics) to GA standard is a very good idea! I just had a look, the article is long, yet in a desperate state. If you want I can leave some suggestions there at the talk page. Jakob.scholbach (talk) 14:43, 24 July 2012 (UTC)

Yes, please do so. I know of several articles that have had quality grade B for a long time but I am not sure how things work when an article evolves tto GA status. Isheden (talk)

Recent edits

I reverted a number of recent edits by Tim Zukas: they are certainly well-intended, but I believe they decrease the article quality: logarithms do not (directly) "involve" exponentiation, they undo it; "base-2 logarithm" is a wording that is not careful enough at this stage of the article, "clearly" is POV that is to be avoided, also "we" does not belong in a WP article. Finally, the article needs to keep the balance between examples and the rest. Jakob.scholbach (talk) 12:05, 14 October 2012 (UTC)

The only objection to examples is they're easy to understand, and that's particularly true of logarithms. Nothing explains logarithms better. Some people think the article shouldn't have too many examples, because that makes things too easy for the reader-- better he should have to struggle to understand. (That makes him a better person.) But if our aim is to help the reader understand, then forget about "balance" between simplicity and obscurity. You can't get simpler than an example, and more examples are simpler (for the reader) than fewer.

Look at the "Definition" section as you prefer it. It defines logarithms that are whole numbers, 2 or greater-- the base-2 log of 8 is 3, it says. It needs to explain logarithms that aren't whole numbers, and the ones that are less than 2, and the ones that are less than zero. (2 to the minus-1 power is 1/2-- why? 2 to the zero power is 1-- why? Because that follows from the log of a product being the sum of the logs of the factors.)

The reader has perhaps noticed that a table of logarithms says the base-10 logarithm of 2 is 0.30103-- what does that mean? The definition section as you prefer it makes no attempt to answer that glaring question.

The ideal Wikipedia article answers the reader's questions as soon as they pop into his head. If you're claiming to "define" logarithms, the reader has the right to hope you'll define all of them. The best way to do that, by far, is with examples.

"logarithms do not (directly) "involve" exponentiation, they undo it"

Actually, logarithms are numbers-- they don't do or undo anything. We're the ones that are doing and undoing. So neither wording actually makes sense, but that's no big deal.Tim Zukas (talk) 22:00, 14 October 2012 (UTC)

Tim, with reference to your comments above, please familiarize yourself with the identified purpose of Wikipedia. You seem to be saying that it must be a textbook, but this is not how it should be. It is understandable that you feel upset, but once you start using math articles for their intended purpose, you will realize that excess examples and explanation detract from this purpose. Jakob has taken care to motivate the removal of your edits; you should also consider that the article did not get to featured article status without a thorough review process involving the consideration of many editors. — Quondum 04:19, 15 October 2012 (UTC)

I entirely agree with Quodum. Tim, saying that 2^0 = 1 because logarithms of products is the sum of logarithms is, at least if you compare to all the literature, going backwards. Instead it is common to define 2^0 := 1 since one wants the product of a sum of exponentials to be the product of the exp's. This is also much easier since the "empty product" is one. Moreover, the article does explain how to calculate logarithms such as log_10 (2), just a bit later in the article. Jakob.scholbach (talk) 10:20, 15 October 2012 (UTC)

"Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter. It is not appropriate to create or edit articles that read as textbooks, with leading questions and systematic problem solutions as examples. These belong on our sister projects, such as Wikibooks, Wikisource, and Wikiversity. Some kinds of examples, specifically those intended to inform rather than to instruct, may be appropriate for inclusion in a Wikipedia article."

If you're informing people what a logarithm is, you can't do better than examples. (If you want your readers to understand what a logarithm is.) "Systematic problem solutions"-- do you think "The base-10 logarithm of the square root of 10 is 0.5" is any such thing?

"the article did not get to featured article status without a thorough review process involving the consideration of many editors."

Talk about unintended consequences-- the people who invented Wikipedia couldn't foresee the effect of "good article" and "featured article" awards. But if they were starting over they'd know better now.

Doubtless there's no hope, but I'll write up the Definition section the way it should be, so it gives the reader a fair idea of how the logarithm of any positive number is defined.Tim Zukas (talk) 19:40, 15 October 2012 (UTC)

Sure, try your hand. For the serenity of all of us, please use the talk page first for a draft. I should also say, I believe your good will serve WP much better if you focus on another article. Really, this article is at a high level (and I am biased saying this, since I carried the article to and through FA), but there are tons of articles, such as, say, exponentiation that deserve our attention more than this one here. This is not to say that this article is perfect (there probably is no such article), but just as a question of probability, it is less likely that an edit improves this article than it will do for others, less well-polished ones.

BTW, the article did receive a great deal of attention. Just look at the sheer length of the FA nomination. Jakob.scholbach (talk) 10:43, 16 October 2012 (UTC)

I've only looked at the Definition section, which is the important one; we'd be too generous to call it middle level. Maybe the article got a billion man-hours of attention, and you figure that means it must be good. That's where you're wrong.Tim Zukas (talk) 19:55, 16 October 2012 (UTC)

In the first place it is never good to judge an article by one section only. The calculation of logarithms is dealt with below. Secondly, I cannot refrain from saying that if this article is below middle level your edits turned it into low level. Jakob.scholbach (talk) 07:45, 17 October 2012 (UTC)

Simple examples

I put some super basic examples in the begining of the article. I feel like someone should be able to look at super basic calculations and learn what a logarithm is without even reading anything. I put the examples log3(9)=2 and such because if someone can just look through and see these super easy quick examples they will learn the object much easier. All math articles should have something like this so that learning becomes instant. Start with the most basic examples then talk about it. This seems like a much smarter approach to things. — Preceding unsigned comment added by 98.164.209.15 (talk) 18:18, 6 November 2012 (UTC)

Binary log in music

I feel something is missing in the table "which use for which logarithm base" and that is the musical field. For example, we can say that two pitches are n octaves apart if and only if n is the binary logarithm of the ratio between the frequencies of the two pitches. Also, two pitches are m cents apart when $m=1200\times \log _{2}\left({\frac {a}{b}}\right)$, a and b being the two frequencies.

This is discussed in detail below in the article. The discussion of "what log applies where" is just a very brief summary of the application sections below. Jakob.scholbach (talk) 15:46, 17 December 2012 (UTC)

Order of magnitude

A mention about order of magnitude should be added to the lead. --Hartz (talk) 06:36, 11 January 2013 (UTC)

Alpha-Log extension

I recently read an article about a generalization of logarithms:

| last=Matsuyama |first=Yasuo | title=The α-EM algorithm: Surrogate likelihood maximization using α-logarithmic information measures | journal=IEEE Transactions on Information Theory | volume=49 | year=2003 |number=3 |pages=692706 .

I am not sure if this article could use an "extensions" section or if alpha-logs should get their own page? Mouse7mouse9 00:12, 17 April 2013 (UTC)

equal

10^n = log(x). 121.7.54.103 (talk) 07:23, 3 May 2013 (UTC)

Why the particular choice of (1 − 10^(−7))^L?

Please refer to the portion of the article: http://en.wikipedia.org/wiki/Logarithm#From_Napier_to_Euler It is not clear why the choice, "... Napier calculated (1 − 10−7)L for L ranging from 1 to 100 ..."? Could you please elaborate in the article? Any source material for that particular choice? bkpsusmitaa 59.93.200.232 (talk) 04:23, 5 September 2013 (UTC)

Question concerning the fundamental property of a logarithm

I know that logarithmic functions have the fundamental property that log(xy) = log(x) + log(y). Does this property work in the other direction? In other words, if a function "f" has the property that f(xy) = f(x) + f(y), is the function definitely logarithmic, or do any other functions that aren't precisely equivalent to the logarithmic function share this property? Also, if so, is there a proof that any function with this property must be logarithmic? — Preceding unsigned comment added by 12.45.169.2 (talk) 14:36, 10 June 2013 (UTC)

Answer

This is a really good question in my opinion.

In his book Foundations of Modern Analysis, item (4.3.1) (at least in my Spanish version), Dieudonné shows the following (which I'm paraphrasing a little bit):

Fix a real nuber a>1. If f:(0,infinity)-> R is a function such that:

i) f is increasing

ii) f(xy)=f(x)+f(y)

iii) f(a)=1

then f is the logarithm to base a (i.e., f is a unique function, continuous, homeomorphic, etc.)

If you want to know what happens to bases 0<a<1, then you can show that if f:(0,infinity)-> R is a function such that:

i') f is decreasing

ii') f(xy)=f(x)+f(y)

iii') f(a)=1

then f is the logarithm to base a;

this is done taking b:=1/a, g:=-f and applying the previous result (since b>1 and g satisfies i)-iii) ).

I've added this information to the Logarithm page in a new subsection called "Characterization of the logarithm function" under the section "Analytic properties". Improvements are welcome!

Jose Brox (talk) 14:43, 17 October 2013 (UTC)

Edited Transcendence of the logarithm

In the section Transcendence of the logarithm, the second paragraph said (9/14/2013): "Complex numbers that are not algebraic are called transcendental;[44] for example, π and e are such numbers. Almost all complex numbers are transcendental. Using these notions, the Gelfond–Scheider theorem states that given two algebraic numbers a and b, logb(a) is either a transcendental number or a rational number p / q (in which case aq = bp, so a and b were closely related to begin with).[45]" I am changing it. If someone wants to expand the section to include logs of complex numbers (which doesn't seem a good idea) then they need to keep real and complex numbers separate. It will be too confusing otherwise. I leave it here in case I screwed it up. The reference [44] supports the first clause in the first sentence, NOT the second (the one that is, I think, wrong (the complex number π+0i is NOT π). The rest of the paragraph is as far as I can see irrelevant and or rubbish. - - I just read that complex numbers are NEVER rational (See Gelfond Scheider), this is further confirmation that this section has been messed up. IMHO, complex logs (and their transcendence) are a different subject72.172.1.28 (talk) 15:20, 14 September 2013 (UTC)

Thank you! It's good that only real numbers are mentioned now, since the complex logarithm is introduced later in the article. Isheden (talk) 13:58, 28 October 2013 (UTC)

Under "Change of Base"....

The "Change of Base" section leads with

"The logarithm logb(x) can be computed from the logarithms of x and b with respect to an arbitrary base k..." and provides an equation that seems invalid when x is zero.

Shouldn't some mention be made that x must be nonzero for the equation to hold, or is this too much from a computational point of view? Jfriedl (talk) 02:07, 14 October 2013 (UTC)

The Definition section above clearly states that x is a positive real number. Isheden (talk) 17:53, 28 October 2013 (UTC)

Recent edits by Jose

Here is a copy of User:Jose Brox's message at my talk page:

---

I was who added the "Characterization of the logarithm function". I do not agree at all with your subsequent change, because:

1) I do not concede that its relative importance is low:

a) Its fundamental property happens to characterize the logarithm even without asking for continuity (just monotonicity), which is important. If you are looking for another function like that (what people try to do from time to time), you are not going to find it. b) Exactly because of this unicity proof, entropy functions (in both Thermodynamics and Information Theory) have to be defined via the logarithm and there is no other kind of function that could be an entropy. (*) c) Dieudonné, arguably one of the best analysts of the 20th Century, chose this result as the opening one for its Logarithms and Exponentials section in his most relevant Analysis book (so to him it was surely not "mostly irrelevant" or secondary).

2) I consider that the subsection where you pasted the result, "Related concepts" under "Generalizations", is totally inappropiate: it is neither a related concept nor a generalization. The correct section for this result is "Analytical properties".

3) Even if you where right about 1) (which I think you are not), I do not consider that the proper order for information in a page is in strictly descending order of importance. Readability and complexity of information are also crucial. Hence I would not agree to paste this result after several ones on limits and integrals, because it is simpler and more elementary than them in its formulation, and because it is a property strictly related with the very definition, so in my opinion it is preferable to have the readers to reach it as close to the definition as possible, even if it a bit less "fundamental" than the following ones.

(*) This could be added to the section or anywhere in the article, and I planned to do it anytime, but I need to look for a good reference to cite and I do not have much time these days.

For the reasons above I am going to revert your change.

Kindest regards! Jose Brox (talk) 11:17, 28 October 2013 (UTC)

---

(end of copy)

Jose, thanks for your message. There are, IMO, a couple of things to consider here: first, the space devoted to some fact / theorem / ... has to be related to the importance for the entire article. As you can see, the topic of log's touches a variety of sciences and centuries. This single theorem, even if Diedonne choses it as his first theorem (which IMO is of little relevance, but that's not important either), is not worth spending about 10% of this article. There is no doubt about that, I think. This is why I trimmed your addition so much. Second, I also trimmed it because it is not in line with the manual of style (things like "Proposition 1" are not WP-style, this is not a text book). Third, about the placement of the sentence (anything else would be too much) carrying the information: you are of course right that the theorem you mention is not a generalization / related concept. However, it is obviously related to the similar theorem about the continuous group isomorphisms. In the interest of space and coherence, I still think this is the best place. I can't think of a better place where you put a single sentence. If you can, I am happy to consider such an alternative. Jakob.scholbach (talk) 16:34, 28 October 2013 (UTC)

A suggested alternative

Jakob, thank you very much for your answer. I agree with the length issue: it could and should be shorter, even a one-liner as you did (although, obviously, I would prefer some elaboration, for example making incidence in that continuity is not a requisite). Which is the problem with having a subsection of "Analytical properties" filled with just one sentece or two? Moreover, I think that how this property affects the possible choices for an entropy function should be also mentioned in the same subsection. Maybe the homomorphism point of view could also be added to that subsection and erased from where it is now. The result could be something like this:

---

Characterization of the logarithm functions

The fundamental property of the logarithm characterizes it [cite Dieudonné's (4.3.1)]:

Fixed two positive real numbers a>1 (a<1) and C, there is only one increasing (decreasing) function from (0,infty) to R such that f(a)=C and f(xy)=f(x)+f(y), namely C·log_a(x). Note that continuity and differentiability are not required, but obtained a posteriori.

This characterization has interesting consequences:

• Since an entropy function (both for Thermodynamics and Information Theory) should satisfy properties [this and that properties, and if the information for indepedent events i and j is I(p_i) and I(p_j) then the information stored in its simultaneous should be I(p_i)+I(p_j)], it turns out that the only way to build an entropy function is by means of the logarithm function.

• From the perspective of abstract algebra [I dislike the "pure mathematics" phrase], the identity log(xy)=log(x)+log(y) expresses a group isomorphism between positive reals under multiplication and reals under addition. By the characterization above, logarithmic functions happen to be the only continuous isomorphisms between these groups, _written in that order_ [otherwise you have the exponential functions]

• [Put here more consequences of this unicity result , if known]

---

What do you think about it?

I apologize for my additions not being in line with Wikipedia's manual of style. I hope I can read it soon. It is the first time I add such a bulk of information to a page, but I really think that the result is important: actually, note that even as it is now, the page is citing it indirectly below, when it says that the logarithms are the only group isomorphisms as stated above (and referring to Bourbaki, which is just Dieudonné with his best friends :P).

Besides all that, Wikipedia being authority-based when we refer to quality and source of information, mentioning Dieudonné's choice of this result is not an empty argument, IMHO.

Regards, Jose Brox (talk) 14:20, 29 October 2013 (UTC)

OK, placing this characterization not as far down in the article is maybe a good idea. I have moved it up a little bit, because the statement does not require knowledge / mention of inverse functions.

I still think the theorem in the form you have given it and the one using continuity is essentially the same, as follows from the (more or less immediate, given that Q is dense in R) exercise: let $f:R\to R$ be an incresing group homomorphism. Then f is the identity (and in particular continuous). Jakob.scholbach (talk) 08:50, 30 October 2013 (UTC)

Alpha logarithm

There is another generalization discussed here: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1184145

$\alpha -Logarithm(\alpha ,x)={\frac {2}{1+\alpha }}\left(r^{\frac {1+\alpha }{2}}-1\right)$

This has the properties:

(i) $\alpha =-1$ is the traditional logarithm

(ii) $\alpha -Logarithm(\alpha ,1)=0$

(iii) it is strictly monotone increasing W.R.T. $x$

(iv) it is strictly concave when $\alpha <1$, a straight line when $\alpha =1$ and strictly convex when $\alpha >1$

More properties listed in the reference — Preceding unsigned comment added by 150.135.223.18 (talk) 21:20, 25 November 2013 (UTC)

products in logs

should point out that:

                log_10(x)+log_10(y)
   log_bc(xy)= ---------------------
                log_10(b)+log_10(c)

i know its similar to what is there, but its only explained you can do it in the power. Charlieb000 (talk) 23:54, 3 June 2014 (UTC)

Simpler page

hey, can someone create a similar page for people like students who are trying to understand the subject. simple english wiki is too simple, and this page explains it, but in a way that is hard to understand if you don't already know what's going on. — Preceding unsigned comment added by KendoSnowman (talk • contribs) 20:07, 26 March 2014 (UTC)

That's a problem on the majority of mathematics and statistics pages on wikipedia. They contain the information but do not do a good job of conveying it. — Preceding unsigned comment added by 89.100.74.53 (talk) 13:38, 22 December 2014 (UTC)

Nomenclature

You've done nothing to show that Napier worked with speed delays, & called it "lag" arithmetic, not log for logos, or logging. Scottish dialect. Peter Sheedy, Email PS20100401@gmx.de95.113.14.40 (talk) 15:18, 3 April 2015 (UTC)

I think I get but I still have some questions...

What's that arrow thingy mean?↔ — Preceding unsigned comment added by 98.118.251.23 (talk) 23:34, 22 April 2015 (UTC)

This notation means "is logically equivalent to". It is unnecessary WP:technical here. So I have edited the article to avoid it (and doing some simplifications and clarifications). D.Lazard (talk) 09:59, 23 April 2015 (UTC)

antilogarithm

Can this pinnacle of Wikipedia explain more clearly that antilogarithm is the same as exponential (as far as I can tell from this article). Unless there is a difference that I'm missing... 86.121.137.79 (talk) 14:07, 28 December 2014 (UTC)

"Antilogarithm" and "exponential function" are sometimes synonymous and sometimes used differently (exponential being only exp(x)). So I think that the explanation in the article is good in that it shows the antilogarithm function explicitly without mentioning the word "exponential".--Jan Spousta (talk) 11:38, 4 March 2015 (UTC)

Reminds me that what used to be an integral is now an antiderivative, at least in many Calculus books. Gah4 (talk) 21:56, 23 April 2015 (UTC)

log+

Hi, log+ (i.e., maximum of 0 and the logarithm). Should it be explained here ? --Adam majewski (talk) 12:32, 18 July 2015 (UTC)

My impulse would be to say "no". max(0, log x)) is not the sort of function that is likely to be treated as a "basic" function in its own right in mathematics, and thus would not be mentioned in an encyclopaedia. Its use is also likely to be related to practicality (implementation) rather than some real mathematical origin. I've never heard of it before. —Quondum 13:43, 18 July 2015 (UTC)

Lede too simple

The lede sentence "In mathematics, the logarithm of a number counts repeated multiplication." is too simplistic, especially as the paragraph after giving a simple example, log(1000)=3, then discusses the logarithm of arbitrary positive real numbers without any explanation or transition. An alternative might be, roughly:

"In mathematics, the logarithm is the inverse operation to exponentiation. In simple cases it counts repeated multiplication. For example... [existing log(1000) example]. More generally exponentiation is defined for any two real numbers, so the logarithm is ..." --agr (talk) 14:58, 17 September 2015 (UTC)

I made the change.--agr (talk) 14:45, 18 September 2015 (UTC)

This change is vague, confusing, and poorly written. "In simple cases it counts repeated multiplication." Multiplication of what, and repeated until when? "For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 10³); the multiplication is repeated three times." The last part should read, "three bases are multiplied." There are actually only two multiplications (and the multiplication is only repeated once). I corrected it and switched it to using repeated division, and included a reference, but it was quickly reverted without discussion with this explanation: "That's a pretty unusual way to describe it; stick with normal." It's "normal" to explain it as an inverse of exponentiation, which is repeated multiplication. It's not normal to write "it counts repeated multiplication," and I could not find any decent source that explains it that way. On the other hand, I did easily find a source that explains it as repeated division (after explaining it the normal way as the inverse of exponentiation). Explaining logarithms as repeated division gives a slightly different perspective that will help people understand. Explaining logarithms as a count of repeated multiplication is basically restating that it is the inverse of exponentiation. If that's the direction you want to take, it should be clearer, like this:

"In mathematics, the logarithm is the inverse operation to exponentiation, which is repeated multiplication of a base number. Thus, in simple cases, the logarithm is the count of how many base numbers need to be multiplied together to equal the operand."

--AndyBloch (talk) 08:16, 6 January 2016 (UTC)

Logarithm chart.svg

Logarithm in base 2-10 of numbers 1-10. Line labels = number. X axis = base. Y axis = logarithm.

What information is this image trying to convey? It claims to show plots of $\log _{x}(1),\log _{x}(2),\log _{x}(3),...,\log _{x}(10)$. Okay, so that's $0,{\frac {\log(2)}{\log(x)}},{\frac {\log(3)}{\log(x)}},...,{\frac {\log(10)}{\log(x)}}$. So it's a plot of $0$ and then nine plots of ${\frac {1}{\log(x)}}$ multiplied by a constant? Further, this plot makes it look like these functions are all piecewise linear, which they are not.

According to the page history this image has been removed at least once already, so I am posting here instead of just removing it myself. — Preceding unsigned comment added by 156.40.252.1 (talk) 18:04, 31 March 2016 (UTC)

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Logarithmic function

This section (like most of the article, I suppose) is in need of drastic improvements, and I'd recommend deleting it unless someone wants to rewrite it completely. It misstates intermediate value theorem , and it misstates the definition of continuous function. I don't see why most of this section is here, since it seems to be reproving that every strictly monotonic function has an inverse function (whose domain is the range of the original function). AndyBloch (talk) 22:06, 2 April 2016 (UTC)

The derivative of ln(x) = 1/x

Would this imaginative proof presented on Stack Exchange, which relies on the fact that x = exp(ln x) and then using the chain rule, be suitable for the general reader of this article? It's the one that starts "If you can use the chain rule and the fact that the derivative," etc.

http://math.stackexchange.com/questions/1341958/proof-of-the-derivative-of-lnx

Meltingpot (talk) 20:10, 29 October 2016 (UTC)

The area under 1/x goes back pretty close to the beginning of calculus. There are books with a good explanation, but pretty much ln(x) was defined is the integral under 1/x, (but not yet named ln(x)) and then later found to be the inverse exponential. Not so obvious the way most books teach it now, but integration was discovered first, and then derivative as its inverse operation, along with the connection to the slope of a curve. There are books that make good references for this. Gah4 (talk) 20:43, 29 October 2016 (UTC)

The fact that the derivative of the log function may be deduced from the fact that x = exp(ln x) by using the chain rule is explicitly described in section "Derivative and antiderivative" of the article. However this require to have first defined the exponential function and the constant e. Therefore a more coherent (but less pedagogical) presentation of mathematics consists to define the logarithm as the antiderivative of 1/x that is zero for x = 1. Then the exponential may be defined as its inverse function or directly, thanks the chain rule, as the unique function which is equal to its derivative, and takes the value 1 at 0. This allows to define e as the exponential of 1. The advantage of such an organization of the definitions is to make clear that there is no circular reasoning. D.Lazard (talk) 08:48, 30 October 2016 (UTC)

Good comments both. D Lazard, you are quite right to say that the section entitled "Derivative and antiderivative" contains the description of the fact I referred to; I should have read the article more carefully before commenting. Meltingpot (talk) 19:12, 4 November 2016 (UTC)

Fractional exponents as discrete logarithms

What is the status of fractional exponents as ordinary discrete logaritms?This could specified in article.--5.2.200.163 (talk) 16:39, 13 September 2017 (UTC)

Footnotes with Google books template

@Purgy Purgatorio: Please notice that right now the templates are not properly closed, so the CS1 template is not properly displayed. Though I suspect {{google books}} might also need some fixing in order to provide a proper input for the CS1 parameter in order for the citation to work well, I am quite certain that at the very least the template calls should be properly closed.- Andrei (talk) 14:59, 23 April 2018 (UTC)

Args in parens

Lately, I observed a run on eliminating parens-pairs enclosing simple atoms as argument of functions denoted by a sequence of tokens, e.g., $\log _{a}$, but never as an argument of a single-token function, e.g., $f$. This witch hunt seems to have reached a boundary within the lede pic, rascally containing $\log _{10}(x)$.

While I fully understand the desire to omit superfluous tokens from a notation, I am by far not thus convinced that in this article on quite basic level the insinuation of versedness in composing maps with or without being explicit wrt arguments is sufficiently enough a reason to omit those functional parens, obviously considered necessary in $f(x)$, which is only quite rarely seen as $fx$ (I do not want to deny the possibility of $x$ having different domains in these cases.)

I am in doubt if the omission of parens here increases the readability of this article, but I do not know about meaningfully applicable rules. Happy reverting? I won't move a finger. Purgy (talk) 06:52, 1 October 2017 (UTC)

This is not only a problem of readability, as the article does not states when parentheses must occur, when they are optional, and when it is better to omit them. Without this this article, and all articles linking to it may be confusing for some readers. My opinion is that parentheses are the norm and they may be ommitted only when no confusion is possible. An other option would be to define a precedence for the log, and adding it to the list of operations in Order of operations. In any case, what should be the rule for $\log a+\log b$, which, theoretically may be read also $\log(a+\log b)?$. I'll edit the lead to clarify this. D.Lazard (talk) 09:08, 1 October 2017 (UTC)

Well, I've heard about a general precedence of unary ops over binary ones, and, to be honest, I do recommend to get used to the look and feel and use of parens pairs, since most of the math apps I met recently require to put arguments of functions within parens, be it sin(.), exp(.), or whatever, but, as said, I'd rather fight for i² = -1, which I do not, either. :) Purgy (talk) 10:44, 1 October 2017 (UTC)

As well as I know, in math books it is usually to see sin x and log x, maybe with the x in an italic math font. But I think f(x), with both f and x in the italic math font. But you need the ()'s for more complicated operands, such as sin(x+y). Gah4 (talk) 14:52, 2 May 2018 (UTC)