Talk:Logarithm/Archive 3

Archive 1

Archive 2

Archive 3

Archive 4

Archive 5

Archive 6

Shortened a subsection

While TeXifying the standalone equations, I shortened a subsection with this edit. I don't think we need the details of this proof here. If it is agreed that the details should be included here after all, then I suggest we do it with a proper source. DVdm (talk) 14:52, 4 December 2010 (UTC)

With this edit I also merged two subsections and removed some more unsourced proofs per wp:NOR (specifically wp:CALC) and wp:NOTTEXTBOOK. DVdm (talk) 17:25, 4 December 2010 (UTC)

GA Review

This review is transcluded from Talk:Logarithm/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: SnottyWong ^gossip 23:21, 15 December 2010 (UTC)

I'm going to start reviewing this article. Should have some comments shortly. SnottyWong ^gossip 23:21, 15 December 2010 (UTC)

Copyedit check

Below are my initial comments on the text itself after reading the entire article, organized by section. The green text are quotes from the article. My comments on the other aspects of the article are below in the "Good Article Criteria" section.

Lead

• By the following formulas, logarithms reduce multiplication to addition and exponentiation to products: - Should "products" be changed to "multiplication"? Add wikilinks to multiplication, addition, and exponentiation.

• Done. Jakob.scholbach (talk) 00:21, 17 December 2010 (UTC)

• The first formula below the lead reads:

$\log _{b}(x\ y)=\log _{b}x+\log _{b}y,\,$

I'm not sure why there is a space between the x and the y, and it confuses the fact that x and y are being multiplied. Shouldn't this be:

$\log _{b}(xy)=\log _{b}x+\log _{b}y,\,$

• Inserted a dot to emphasize multiplication. Jakob.scholbach (talk) 00:21, 17 December 2010 (UTC)
• no need for a dot or space in a simple product like this.--JohnBlackburne^words_deeds 13:55, 18 December 2010 (UTC)

• The logarithm to base b = 10 is called common logarithm, - this should be the common logarithm, or a common logarithm. Missing the article.
  • Done. Jakob.scholbach (talk) 00:27, 17 December 2010 (UTC)
    • This now reads The common logarithm has base b = 10 and is most often used for calculation. What does that mean? Calculation of what? SnottyWong ^chat 17:12, 21 December 2010 (UTC)
      • I've clarified it: the previous formulation was confusing as, as far as I know, they are no longer used.--JohnBlackburne^words_deeds 17:24, 21 December 2010 (UTC)

I clarified it some more. I think it was unclear as stated. Dicklyon (talk) 22:01, 27 December 2010 (UTC)

• The invention of logarithms is due to John Napier in the early 17th century. This sentence is strangely worded. Also, are mathematical concepts really "invented"? Did someone "invent" addition? Perhaps "discovery" or "development" would be a better choice of wording here.

• Replaced by "development". Jakob.scholbach (talk) 00:21, 17 December 2010 (UTC)

It's better to stick to sources, and say he invented logarithms. He invented them as a computational aid; the mathematical function concept was developed later. Dicklyon (talk) 21:39, 27 December 2010 (UTC)

• Before calculators became available, via logarithm tables, logarithms were crucial to simplifying scientific calculations. If you take out the clause in the middle of this sentence, it reads "Before calculators became available, logarithms were crucial to simplifying scientific calculations." That makes no sense. What happened after calculators became available? Were logarithms any less crucial?

• hopefully fixed as part of general ce of last two paragraphs.--JohnBlackburne^words_deeds 13:55, 18 December 2010 (UTC)

• In addition to being a standard function used in various scientific formulas, logarithms appear in determining the complexity of algorithms and of fractals. "Appear" should probably be changed to "are used".

• OK. Jakob.scholbach (talk) 00:21, 17 December 2010 (UTC)

• The last two paragraphs of the lead are choppy and disjointed, and are difficult to read. Careful copyediting for cohesion would be beneficial here.

• I've copy-edited the last two paragraphs, rewriting maybe half the text and rearranging everything.--JohnBlackburne^words_deeds 13:55, 18 December 2010 (UTC)

Logarithm of positive real numbers

• For b = 2, for example, this means $\log _{2}(8)=3,\,$ - Shouldn't this read, "For b=2 and y=8, for example, this means..."
  • Sure. Jakob.scholbach (talk) 00:25, 17 December 2010 (UTC)

• Ways of calculating the logarithm are explained further down. - Could be less informal. Perhaps replace with something like "Methods of calculating the logarithm can be found in the Calculations section of this article."
  • OK. Jakob.scholbach (talk) 00:25, 17 December 2010 (UTC)

• The logarithm logb(y) is defined for any positive number y and any positive base b which is unequal to 1. These restrictions are explained below. This sentence could be deleted and simply explained in more detail in the appropriate section below. This section has a lot of links to other sections in the same article, which is odd. It's almost acting as a "second lead", directing the reader to various sections within the article. I don't think this is necessary or desired.
  • I removed one internal link. For the remaining one: I think we need to have up front the information what numbers the log is defined for. Otherwise it would just be incomplete and therefore partially wrong. However, justifying this is kind of less interesting for some readers and anyway seems to fit most naturally in the actual proof of well-definedness of the logs given in section 4.1. Jakob.scholbach (talk) 02:05, 19 December 2010 (UTC)
    • Agreed. SnottyWong ^chatter 17:12, 21 December 2010 (UTC)

• The name of this section isn't ideal. Being that it is the first section after the lead, I would name it something more general like "Overview" or "Overview of Logarithms". The section itself doesn't really mention anything about positive real numbers. The section as a whole also is a little bit on the WP:HOWTO side, although I think it's acceptable as is for now, and may even be a good thing for a technical article. This may need to be reviewed if the article is to progress to FA status.
  • Hm. In principle Overview is a nice name for a first section, but here it seems inappropriate to me, given that this section is just not an overview. I'm open to further suggestions, but I think we have to stick to section titles which do summarize the contents of the respective section. About HOWTO: having the examples is certainly a must for such an article, removing them would render the article useless for many readers. Jakob.scholbach (talk) 02:05, 19 December 2010 (UTC)
    • Agreed. SnottyWong ^chatter 17:12, 21 December 2010 (UTC)

Logarithmic identities

• The first is about the logarithm of a product, the second about logarithms of powers and the third involves logarithms with respect to different bases. - This sentence is entirely unnecessary and can be deleted.
  • OK. Jakob.scholbach (talk) 07:58, 17 December 2010 (UTC)
    • Rereading this section again, I noticed it starts out with an introductory sentence which says that there are three important formulas for logarithms, followed by four formulas. Are there three important formulas or four? It seems that the fourth formula (log of a root) is exactly the same as the third formula (log of an exponent), since a root is just reciprocal exponent. I'm going to attempt to fix this one myself, let me know if you have any problems with my fix. SnottyWong ^{spill the beans} 18:30, 21 December 2010 (UTC)
      • It's probably not a good idea to "fix" that. Most sources teach it this way (e.g. this book or this book); for people not familiar with a root being the same as a power, this is a way for them to see that. Dicklyon (talk) 20:04, 21 December 2010 (UTC)
        • Fair enough. Feel free to revert my changes if you like. However, either way, something needs to be fixed. Either the intro to the section needs to say that there are four important formulas, or one of the formulas needs to be deleted so that the statement that there are three important formulas is correct. Looking at Logarithmic identities#Using simpler operations, it looks like there are actually six important formulas? SnottyWong ^comment 20:51, 21 December 2010 (UTC)

• Subsection "Logarithm of product, quotient, power and root" could be more succinctly renamed to "Algebraic identities".
  • "Algebraic identities" is unspecific and tells little to people not knowing the notion of algebra, so I prefer keeping the current title. Jakob.scholbach (talk) 07:58, 17 December 2010 (UTC)

• The following formula relates the logarithm of a fixed number x to one base in terms of the one to another base: - This sentence is strangely worded and not entirely comprehensible.
  • Reworded. Jakob.scholbach (talk) 07:58, 17 December 2010 (UTC)

• As a practical consequence, logarithms with respect to any base k can be calculated with a calculator, when logarithms to any base b (often b = 10 or b = e) are available: - I understand what this sentence means because I am familiar with logarithms, but I fear someone who is not familiar will not understand. I think this sentence needs to make it clear that it is talking about situation when a certain base is not available on a calculator. Possible rewording could be: "As a practical consequence, logarithms to any base can be calculated with a calculator, even if the logarithm function to that base is not available on the calculator. For instance, to calculate a logarithm to base k with a calculator which can only calculate logarithms to base b:"

$\log _{k}x={\frac {\log _{b}x}{\log _{b}k}}.\,$

• • Reworked. Jakob.scholbach (talk) 07:58, 17 December 2010 (UTC)

• In the two formulas under the subheading of "Change of base", the first formula is solving for the logarithm of base b, and the second formula is solving for a logarithm of base k. For consistency, the left side of the equation should always be solving for the logarithm of base b, or else things get confusing.
  • Right. Jakob.scholbach (talk) 07:58, 17 December 2010 (UTC)

Particular bases

• Given a number n and its logarithm log_b(n), the base b can be determined by the following formula:

$b=n^{\frac {1}{\log _{b}(n)}}.$

This follows from the change-of-base formula above. - I'm unsure why this statement and formula appears in this section, and what relevance it has to the table below it.

• • Moved up to base-change and reworded. Jakob.scholbach (talk) 08:06, 17 December 2010 (UTC)

• Within the table, the parenthetical statement: (in mathematics and many programming languages including C, Java, Haskell, and BASIC) is making middle column of the table very wide. Perhaps the statement could be limited to "(in mathematics and many programming languages)", or it could be wrapped onto another line, moved to a footnote, or moved to the "Used in" column.
  • OK, now in a footnote. Jakob.scholbach (talk) 08:06, 17 December 2010 (UTC)

• Within the table, we have the parenthetical statement (see decibel and see below), again directing us to another section in the article. This should be shortened to "(see Decibel)".
  • Why? (This is a general point where we seem to disagree...) Jakob.scholbach (talk) 01:58, 19 December 2010 (UTC)
    • This may just be a personal preference of mine. I took a quick look through the MOS but couldn't find a passage which discourages (or encourages) the frequent use of section links. I guess I don't find them helpful. Just saying "see below" doesn't direct the reader to any specific area of the article, and therefore isn't helpful. All it's saying is "if you're confused, keep reading and your questions might be answered later." Linking to a section is a bit more helpful, but if anyone ever changes the section name, the link will be broken (and the person who changed the section name will have no way of knowing that they broke a link). I'm ok with the reference to the decibel section, but I still don't see the need or use of "see below".

Analytic properties

• The following discussion of logarithms uses the notion of function. In a nutshell, a function is a datum that assigns to a given number another number. The first one is called variable to emphasize the idea that it can take different values. - A couple of things. First of all, any reader that has made it this far in the article probably knows what a function is already. If they don't, they're probably not going to understand the rest of the article. Second of all, the definition given for a function is extremely confusing. I understand the concept of a function but if I were explaining it to my grandmother, I wouldn't say that it is a datum that assigns to a given number another number. I think this entire statement should be removed.
  • I disagree with your first statement. Logarithms can and usually are defined at middle/high school level as we do it in the first section. The notion of function is typically introduced later in the curriculum. This group of readers will benefit from a careful introduction of the concepts. The same holds true for continuity and differentiability. This is why I gave a short explanation of these notions and I would not want to remove them just because we have of course more detailed articles. Continuous function, e.g. welcomes the reader with a cleanup tag, a choppy lead and an equally choppy main article. I fail to see how the log. article becomes better by removing the respective explanations.
  • OK, so how would you explain it to your grandmother? Jakob.scholbach (talk) 08:35, 17 December 2010 (UTC)
    • That's a tough (maybe impossible) concept to explain to your grandmother in one or two sentences. That's why I would favor linking to Function (mathematics), and if someone is truly unfamiliar with what a function is, they can learn there. If we're assuming our reader does not know what a function is, I don't think it's possible to sufficiently explain the concept to them in one or two sentences. The first two sentences of Function (mathematics) are a good start though: "The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type." SnottyWong ^babble 19:50, 21 December 2010 (UTC)
• The expression logb(x) depends on both the base b and on x. The base b is usually regarded as fixed. Therefore the logarithm only depends on the variable x, a positive real number. Assigning to x its logarithm logb(x) therefore is a function. It is called logarithm function or logarithmic function or even just logarithm. - This is probably the third time in this article that the logarithm has been defined at a very basic level. I really don't think any of this is necessary at this point, and the wording is very choppy and disjointed. I would suggest deleting it and rewording the following sentence to say "Logarithms can be defined indirectly by means of the exponential function..."
  • This is now reorganized. Please check again. Jakob.scholbach (talk) 01:07, 19 December 2010 (UTC)

• A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen. - This information can be found at Continuous function, which is why you linked to it in the previous sentence. Delete this sentence.
  • I disagree, see above. Jakob.scholbach (talk) 08:35, 17 December 2010 (UTC)

• Moreover, it takes arbitrarily big and arbitrarily small positive values, so that any number y can be boxed by y0 and y1 which are values of the function f. - I'm unfamiliar with the term "boxed" in this context. This term should either be changed to something more common, defined more clearly, or linked to an article which defines it more clearly.
  • Reworked. Jakob.scholbach (talk) 08:35, 17 December 2010 (UTC)

• Roughly speaking, a differentiable function is one whose graph has no sharp "corners". - This information can be found at Differentiable function, which is why you linked to it in the previous sentence. Delete this sentence.
  • I disagree, see above. Jakob.scholbach (talk) 08:35, 17 December 2010 (UTC)

• This can be derived from the definition as the inverse function of ex, using the chain rule. - Grammar is not making sense in this sentence. Do you mean "the definition of"?
  • Simplified the sentence structure. Jakob.scholbach (talk) 01:07, 19 December 2010 (UTC)

• Since this section discusses logarithms with respect to calculus, would it not be relevant to also have some discussion of the derivatives/integrals of logarithms to bases other than e? Why is this section strictly limited to natural logs?
  • It does mention it at the end of the section. More detailed formulas are found at the (linked) article List of integrals of logarithmic functions. I think these variants are not important enough to justify separate treatment here. Jakob.scholbach (talk) 01:07, 19 December 2010 (UTC)

• The figure with caption "A visual proof of the product formula of the natural logarithm." is messing up the formatting of the page because it is center-justified. This should probably be right-justified.
  • How exactly is it messing up the layout? It looks rather nice for me and I put it this way because the image is very large, so if it would be right-centered the remaining space for text would be unusually small. However, if you prefer, feel free to change it. Jakob.scholbach (talk) 01:07, 19 December 2010 (UTC)

• This relation is used in the performance analysis of algorithms such as quicksort, see below. - No need for the "see below".
  • Why not? Jakob.scholbach (talk) 01:07, 19 December 2010 (UTC)

Calculation

• One method uses power series, that is to say a sequence of polynomials whose values get arbitrarily close to the exact value of the logarithm. - Missing an article. Should be "One method uses the power series", or "..a power series".
  • Was actually supposed to be plural. Tweaked. Jakob.scholbach (talk) 09:05, 17 December 2010 (UTC)

• Finally, based on quick ways to calculate exponentials e^y, the natural logarithm of x, that is the solution a to e^a - x = 0 can also efficiently be calculated using Newton's method. Not fully understanding this sentence. I think there are too many clauses.
  • Trimmed. Jakob.scholbach (talk) 09:05, 17 December 2010 (UTC)

• While in some cases—such as log₁₀(10.000) = 4—logarithms can be easy to compute... - The long dashes here are confusing. Also, per WP:MOSNUM, the period should be a comma, i.e. it should be log₁₀(10,000) = 4.
  • OK. Jakob.scholbach (talk) 09:05, 17 December 2010 (UTC)

• ...they generally take less simple values: by the Gelfond–Schneider theorem, given two algebraic numbers a and b such as ${\sqrt[{3}]{2}}$ or ${\sqrt {-5+{\sqrt {3/13}}}}$, the ratio γ = ln(a) / ln(b) is either a rational number p / q (in which case a^q = b^p) or transcendental. - I'm not understanding this sentence at all. What does the cube root of 2 or the long radical have to do with anything? The sentence started out describing how some logarithms are easy to calculate, and ended by saying that the ratio of two natural logarithms is either rational or transcendental. This seems like a lot of random information crammed into one sentence, and there is no discernible relationship between the information.
  • Tried to improve this. Please double-check. Jakob.scholbach (talk) 00:45, 19 December 2010 (UTC)

• Here M denotes the arithmetic-geometric mean and m is chosen so that s = x / 2^m is bigger than 2^(p/2). - Not understanding this. We have to choose m such that an equation is bigger than 2^(p/2)? What has to be bigger than 2^(p/2)? s? m? It doesn't make sense to say that an entire equation needs to be "bigger" than a particular value. This needs to be clarified.
  • Reorganized. Jakob.scholbach (talk) 00:44, 17 December 2010 (UTC)

• The constants π and ln(2) can be calculated with particular series. Which series? Is it even necessary to mention this?
  • Well, there are tons of series calculating pi. I tried a rewording to make it better. Jakob.scholbach (talk) 09:05, 17 December 2010 (UTC)

• The formula goes back to Gauss. Unencyclopedic sentence. Remove or modify.
  • Reworded. Jakob.scholbach (talk) 00:44, 17 December 2010 (UTC)

Complex logarithm

• Consequently, if φ is the principal argument Arg(z), the number number - Typo. "Number" appears twice unnecessarily.
  • Goodness. Now fixed. Jakob.scholbach (talk) 09:09, 17 December 2010 (UTC)

• Accordingly, a is called complex logarithm of z. - Missing an article. "...a is called the complex logarithm of z."
  • Done. Jakob.scholbach (talk) 09:09, 17 December 2010 (UTC)

• If n = 0, a is called principal value of the logarithm, denoted Log(z). - Missing article.
  • Done. Jakob.scholbach (talk) 09:09, 17 December 2010 (UTC)

• ...hence the principal logarithm of such a number is a real number and equals the natural logarithm as defined above. - The word "above" has a link which goes to a non-existent section of this article. The link should probably be removed (as should the text "as defined above") rather than correcting the link.
  • OK, removed. Jakob.scholbach (talk) 00:26, 18 December 2010 (UTC)

• In contrast to the real case, analogous formula for principal values of logarithm of products and powers for complex numbers do in general not hold. - A bizarre sentence. Obvious number agreement problems and missing articles. Also, "do in general not hold" is a wikilink to Exponentiation. I don't understand the significance of that link, and exponentiation has been linked already (probably several times) in the article, so the link should be removed (and, if appropriate, the significance of exponentiation with regard to this statement could be elaborated upon in prose).
  • I don't see the point in removing the link. It links specifically at the section of Exponentiation explaining this. You are right that the material of the linked section might also be put to complex logarithm, but currently it is not and at this point I'm not up to reorganizing the exp. and cx. log. articles. Jakob.scholbach (talk) 00:26, 18 December 2010 (UTC)

• In general, this section is very well-worded (especially the top two-thirds of it) and should serve as a model for how the other technical sections of this article should appear.

Uses and occurences

• I.e., the amount of hard disk space on a computer grows logarithmically as a function of the size of the number to store. Didn't quite understand this example. Hard disk space increases as a function of the size of a stored number? Needs clarification.
  • Reworded. Jakob.scholbach (talk) 00:41, 18 December 2010 (UTC)

• For any given number x, the number of prime numbers less than or equal to x is denoted π(x). - Using the word "number" a lot here. Might be better to change this to "For any given number x, the quantity of prime numbers..."
  • OK. Jakob.scholbach (talk) 00:41, 18 December 2010 (UTC)

• Logarithms appear in the encoding of musical tones. - "Encoding" is a strange choice of word. Musical tones are not encoded. Maybe change "encoding" to "frequencies".
  • Reworded. Jakob.scholbach (talk) 00:41, 18 December 2010 (UTC)

• ...the interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio. - Strange wording. Should be "...the interval between two semitones is..."
  • Semitones is the unit of the measurement so to say. Like "the height of the tree in yards is ...". I currently don't find a better wording for this. Jakob.scholbach (talk) 00:41, 18 December 2010 (UTC)
  • This is now reworked. Jakob.scholbach (talk) 11:10, 19 December 2010 (UTC)

Related notions

• The logarithm of a matrix is the inverse function of the matrix exponential. This lone sentence should at least have a minimal explanation accompanying it about what relevance this has, why it is important, and/or how this is used in mathematics.
  • It is now restructured so as to make it more smooth. Jakob.scholbach (talk) 01:01, 18 December 2010 (UTC)

History

• Napier first called L "artificial number"... - Missing an article. Napier first called L the artificial number, or an artificial number?
  • OK. Jakob.scholbach (talk) 01:56, 19 December 2010 (UTC)

• The calculation of products using logarithms stakes on the following formula: - "Stakes" is a strange verb to use here. Do you mean "depends"? Or maybe replace "stakes on" with "requires"? Some rewording is required here. It's also very unclear what this whole section (starting from this sentence down to the sentence that starts with "For different needs") has to do with the history of logarithms.
  • I tried to make it clear that log tables are an historical application/tool of logs, which is why I think they belong here (as opposed to "Uses and occurrences"). Jakob.scholbach (talk) 01:56, 19 December 2010 (UTC)

• Why is the history section at the bottom of the article? Traditionally, the history section comes right after the lead, and it would seem more appropriate in that location. Is there a particular reason that is not the case, or did it just end up that way? The overall order of the sections in the article should be re-evaluated as well. I would vote for History, then Uses & Occurences, and then all of the more technical sections after that.
  • I don't think it should come immediately after the lead, because at this point we don't even know what logs are. Also, the formulas used in the history section come up in section 2, so history is bound to appear after that. The only other place I could imagine for history is Related notions. I don't have a strong feeling about switching these two sections, but since history seems to round off the article, I slightly prefer it at the very end. For similar reasons, I would oppose moving Uses and Occurrences too far up. We have to give the reader a chance of understanding the notation etc. used there by putting the sections introducing them before. Jakob.scholbach (talk) 01:56, 19 December 2010 (UTC)
    • Fair enough. Again, this is probably just a personal preference of mine. I see some of the other technical math articles do put the history at the end, so there may be a precedent for this. SnottyWong ^speak 20:39, 21 December 2010 (UTC)

Good Article Criteria

I will evaluate the article based on the good article criteria listed at WP:GACR and list the results below:

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS for lead, layout, word choice, fiction, and lists):

   There are numerous grammar problems, and some MOS problems, as noted in the detailed comments above. Overall, the article does not read well, consecutive sentences often don't have a relation to each other, some topics are discussed multiple times in the article, etc.

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

   Article does occasionally stray into the territory of unnecessary detail, but not egregiously so.

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars, etc.:

6. It is illustrated by images, where possible and appropriate.

   a (images are tagged and non-free images have fair use rationales): b (appropriate use with suitable captions):

   The article has quite a lot of images, some would say too many. The overall formatting of the images is not consistent throughout the article, resulting in a messy look. It would be worthwhile to evaluate the images in this article and delete the unnecessary ones.

7. Overall:

   Pass/Fail:

   This is a vital article on a very important subject in mathematics, and it's coming along nicely. However, it is not yet up to GA standards. The majority of the problems have to do with copyediting (as evidenced by the volume of specific comments above), however there are some other problems with the structure and content of the article. The article still needs major work, so I don't see any value in putting the GA nomination on hold. I'd encourage you to overhaul the article using some of the suggestions above, and then nominate the article again. Feel free to contact me on my talk page with any questions. Thanks. SnottyWong ^express 20:52, 16 December 2010 (UTC)

Thanks, Snottywong for the review! All detailed points above have been dealt with (most of them by following your suggestions). An entire overhaul with respect to wording, cohesion etc. of the article will follow. I have also reevaluated the pictures and removed two of them. I'm kind of disagreeing with your point that internal links should be avoided, either by simply removing them or by organising the article such that they become unnecessary. The former just removes a useful bit of information which to keep comes at virtually no cost. The latter: for a broad topic like this, the knowledge about it usually does not come linearly, i.e. a implies b implies c etc. On the contrary, facts are often (historically or factually) intertwined with each other, and I don't see the point in flattening this ontology just for its own sake. That said, I'm happy to consider more specific suggestions how to reorganize the article. Currently, though, I fail to see what major work on the content and structure of the article is needed, so I'd appreciate more specific feedback about this. Thanks again, Jakob.scholbach (talk) 02:16, 19 December 2010 (UTC)

Thanks for addressing my comments above. I stopped my review at the copyediting problems, because there were so many of them and I wasn't sure if anyone was even going to address those comments. I think you have addressed most of the points I brought up satisfactorily, however the article still has some ways to go before it will pass as a GA. Here are a couple of larger issues to address with the article before renominating it:

• There are likely some copyediting issues that I've missed. I was going to recommend you list the article at WP:GOCE/REQ, but it appears you've already done that, which is great.
• Wikipedia articles are written by multiple editors, often with little or no coordination between them. The result often becomes a mish-mash of topics with poor cohesion or a "basket of sentences" that have little or no relation to one another, and no logical transition from one to the next. Now that you've gotten all or most of the small copyedit problems (i.e. single-word or single-sentence problems) corrected, it's time to look at the bigger picture. I think that what this article needs is a "meta-copyedit", a process which will rearrange and modify the material in the article such that it reads like it was written by one person. Information needs to be presented in a logical order, with transitions between different subjects. Random interjections of information need to be woven into the article so that they are no longer random interjections.
• The History section in particular needs a lot of work. Use articles like Integral#History and Calculus#History as examples of how this section should be structured. The History section needs to read like a narrative, telling the story of the logarithm from its origin to present times, with a strong sense of the chronology. The section currently is riddled with formulas and equations and various overly-specific trivia like the derivation of how Napier calculated 10⁷(1 − 10⁻⁷)^L, etc. The section also has several one sentence paragraphs that have no relation to the other material (i.e. the random interjections mentioned above), and are not woven into a narrative about the history of the logarithm. For instance, The work of Cavalieri (Italy), Wingate (France), Fengzuo (China), and Kepler's Chilias logarithmorum (Germany) helped spread the concept further. This sentence should be expanded into multiple paragraphs, showing specifically how their work helped spread the concept further. Overall, the section starts off well, talking about the origin of the logarithms with Virasena in the 8th century, but then jumps to 1544 (nothing happened in between?), then jumps to John Napier and gets into an overly specific description of his work, and then we lose a sense of time altogether and start talking about log tables and slide rules.

—SnottyWong ^chatter 20:33, 21 December 2010 (UTC)

From studying the historical references, it seems that the history of logarithms is by and large incomparable to the one of, say, calculus. Basically, logarithms appeared "over night" (or, after Napier calculating them for 20 years). There is not so much to tell. In particular, no reference notes anything between the "precursors" (most don't even mention them) and Napier/Bürgi. Also, the influence of the four mathematicians your quote refers to is barely mentioned in the books, so I don't know how to write a whole section about this. I'll think about a more historically engaging way of telling the stuff that is to be told, but we can't just invent things. The more I think about it, the more I'm convinced that your expectations in this respect are neither part of GA criteria and nor satisfiable independently of the criteria. Jakob.scholbach (talk) 21:08, 26 December 2010 (UTC)

I disagree. The GA criteria require (among other things) that an article be written in a clear and concise manner, with cohesive paragraphs, and without spelling and grammar errors. It also requires that the article is broad in its coverage, addressing all of the important points without straying into unnecessary details. I believe that portions of this article continue to fail these criteria, per my comments above. You may have a point that there isn't much more of a story to tell in the history section; I don't pretend to have studied the sources in great enough detail to argue. However, I still believe that what story there is to tell must be told better. Excessive derivations and equations don't have a place in the history section of the article, in my opinion. I think we've made a good amount of progress in this process, and there is still some to go. I would encourage you to continue making improvements, and the renominate for GA. If you believe the article currently satisfies the GA criteria and you disagree with my assessment, you may also apply for reassessment. SnottyWong ^{spill the beans} 18:56, 27 December 2010 (UTC)

Copyedits

My copyedits are complete, although I'm happy to repair any damage I created! A few notes:

• Not sure why the Gelford Schneider stuff is in there, but I left it.
• I replaced Complexity with Computing as a section heading only because the material covered more than complexity theory. Another editor reverted it, and I left it that way.
• Many of my changes fall under the general category of removing phrasing that was about the article or the article's phrasing rather than about the subject. This is the main way that the word count fell from 5102 to 4711.
• I struggled somewhat with Logarithm#Logarithm as a function. I'd appreciate it if someone would verify its correctness.

Thanks for your patience while I worked through this essential article. Lfstevens (talk) 16:58, 7 January 2011 (UTC)

You did an excellent job! DVdm (talk) 18:05, 7 January 2011 (UTC)

Yes, many thanks! I tend to work reactively, and I hope that where your edits called my attention to something I did a good thing. Others are invited to review. Dicklyon (talk) 18:19, 7 January 2011 (UTC)

There's just two little things that I would do differently (and always do on other articles):

1. I propose we move inline citations from standalone equations to inline text immediately preceding the equations. To show what I mean, I have made this change. If nobody minds, I'll do the few more later or tommorow...
2. I notice that most but not all equations have no terminating punctuation anymore, so to technically, the sentences in which they live are somewhat crippled now. I don't know what is normal (standard) practice, but it looks like most of our equations do have punctuation.

DVdm (talk) 18:38, 7 January 2011 (UTC)

I've been watching impressed by every edit I looked at, sometimes amazed at the improvement: it's easy to overlook such problems when you're reading and looking at the math as I've been mostly doing. So, yes, excellent work! To DVdm: yes, punctuation after formulae is correct, as per MOS:MATH#Punctuation after formulae.--JohnBlackburne^words_deeds 19:15, 7 January 2011 (UTC)

I deleted the punctuation. I'll restore it. Lfstevens (talk) 23:08, 7 January 2011 (UTC)

OK, great job. Since nobody seemed to object, I also moved 3 more citations per my previous comment. Cheers - DVdm (talk) 14:36, 8 January 2011 (UTC)

Thanks, Lfstevens for your work! I checked the changes. Among a few minor tweaks I undid one change, the separation of the Applications into Outside mathematics and inside: the resulting Mathematics section had only two subsections, one of which was stubby (and more closely related to the Complexity aspects, so I moved it there). Hence only the number theory section remained.

I'm nominating the article for GA again, since I think the failed nomination showed that the article had copyediting problems, which are now (hopefully!) solved. The issue brought up in the GA nomination with the History section is, IMO, unfounded and so should not be seen as an obstacle to a second nomination. Jakob.scholbach (talk) 19:12, 9 January 2011 (UTC)

GA Review

This review is transcluded from Talk:Logarithm/GA2. The edit link for this section can be used to add comments to the review.

Reviewer: Racepacket (talk) 03:28, 17 January 2011 (UTC)

Please fix disamb link: Random number and Shell. No bad links.

Done. Jakob.scholbach (talk) 22:35, 17 January 2011 (UTC)

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS for lead, layout, word choice, fiction, and lists):

   Please consider whether the lead can be improved? Would a math novice get by with just reading the lead and skipping the rest of the article?

   This has been subject to repeated attention (most recently by Lfstevens), but I'll revisit it! Any particular suggestions/concerns? Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   I tweaked it a little bit. We do have things like complex number, inverse function, fractals, which will be difficult to comprehend for, say, a 15y old. However, I feel that the points which are elementary are described in an elementary way. Areas which are accessible only with more mathematical (or other, say for forensic accounting) background or curiosity are presented in a manner that does not try to dumb it down. All in all I think this is a reasonably self-contained lead section. Do you agree? (If not, what could be done better?) Jakob.scholbach (talk) 22:48, 18 January 2011 (UTC)

   "Write the logarithm of x to the base b as logb(x), such as log10(1000) = 3, using parentheses for clarity." - avoid imparative mode. How about, "People write..." Are the parentheses for "clarity" or a result of generalized function notation?

   Done (and removed the parenthesis point). Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   "They describe musical intervals, inform" - do you mean logarithms or logarithmic scales?

   Good catch. Now rearranged. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   I suggest that you use a method other than {{cref}} The notation is not widely-used and is so small it is hard to read. use group ref feature instead - it designates in numerical or alphabetical order.

   Done. Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   "Derivatives and antiderivatives of logarithms to other bases can be derived therefrom using the formula for change of bases." -> "Derivatives and antiderivatives of logarithms to other bases can be derived from this equation using the formula for change of bases."

   Good idea. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   "or transcendental.[12]"->"or a transcendental number.[12]" - parallel sentence construction.

   OK. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   "putting A = z/exp(y),"- would this be more readable (and more consistent with the next equation) using <math>A = \frac(z)(exp(y))</math>, instead of an inline fraction?

   OK. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   Please rephrase: "Logarithms are used in the process of maximum likelihood estimation when applied to a sample consisting of independent random variables: maximizing the product of the random variables is equivalent to maximizing the logarithm of the product, differentiating a sum rather than a product." - the word diffentiating has multiple meanings.

   Done (removed the "differentiation"). Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   Please elaborate that a slide rule is based on its scales being marked in logarithmic distances.

   Done. (Along with the reorganization of the pictures). Jakob.scholbach (talk) 22:49, 18 January 2011 (UTC)

   "Mathematically unsophisticated individuals"->"Psychological experments find that mathematically unsophisticated individuals"

   I don't have strong feelings about this, but why do you want this reword? Your suggested addition makes it slightly longer and does not seem to convey much more information (IMO)? The other reason, taking a more distant point of view than by merely stating this fact, also seems not to be necessary here. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   How about "Studies show that math..." I am worried that you are popping off an interesting fact without relating it to the world of psychological research. The reader will ask how do you know this and why is this sentence here?

   OK, that's a fair point. Now reworded. Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   "The logarithm of a log-normal distribution is normally distributed.[36]" - can you give a real world example of a log-normal distribution so that we can avoid a one sentence paragraph?

   OK, I gave an interesting one (even though I feel kind of incompetent to judge what example is most relevant.) Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

   In history section, how about adding the use in computing the Ballistic Trajectory tables of artillery shells in World War I. (In World War II, they used the first digital computers to compute the tables.)

   I'm not in principle against, but none of the sources I saw mentioned this particular application. Therefore I did not (and still would not) include it here. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   Ordinance computations was a big computational consumer and financed the first computer. See: [1] and Computers: A Life Story of a Technology.

   I don't doubt at all that ballistic computations motivated building computers. But I fail to see how this is related to logarithms. Do you know this? The links you provide don't seem to back up that close a relation. (The first doesn't seem to mention log's at all, the second mentions the relation of logarithms to the slide rule (which we do cover) but the few pages I can see in google don't seem to talk about ballistic curves)? If you can provide a specific source that talks about the connection of log's (or log tables, as opposed to other mathematical tables) and ballistic curves I'm open to include it here, but right now I think they are not closely related or unrelated. Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   Why do you include an example of an equation for a straight line of a semi-log graph paper, but did not include an example of a straight line for log-log graph paper?

   Again, I'm open to discussion here. I did not include it because the (few) sources dealing with log graphs mostly focus on log-linear graphs, less so on log-log graphs. Maybe because the former reduce the large-scaled exponential functions (as opposed to polynomials) to straight lines? Whatever the reason is, I think this article should for balance reasons only contain one of the two possible pictures, and I think the log-linear is more important and also easier to understand. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   Without actually displaying the log-log graph, couldn't you give an example (either as an equation or as a real world application) where someone would want to draw a log-log graph and would expect a straight line.

   I briefly explained how they work and linked to power laws. Any more than this should go, I believe, to the (quite stubby!) sub-article. Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   When I was in 9th grade, we were trained to use both a slide rule and also logarithm tables. In order to use log tables we were introduced to the concepts of characteristic and mantissa. These concepts later helped me understand scientific notation and computer internal representations of floating point numbers. (Now called exponent and significand.) While I agree that "characteristic" and "mantissa" have no theoretical significance, an argument can be made that they should be mentioned in the article. Otherwise, readers may wonder why the C/D scale on your photo of the slide rule only goes from 1 to 10 or how a log table could fit in the back of a math textbook.

   I provided a brief explanation of these notions next to the log tables. Jakob.scholbach (talk) 21:38, 19 January 2011 (UTC)

   Please consider using the image File:Slide rule example2.svg in addition to the photo.

   Thanks for this. I think I will create a tiny variant of this file showing a particular point highlighted and replace the photo by this svg file. However, this might take a few days. Jakob.scholbach (talk) 22:32, 17 January 2011 (UTC)

   Please keep the photo of the slide rule, perhaps at a thumbnail size. Before you create more illustrations, look at what is being used in the article Slide rule and Common logarithm.

   I did replace the photograph. I hope this is OK with you? Here is why: the photo was of course more realistic, but hard to see what is going on because it is quite small (even at the size we had it). We do not (and cannot, for space reasons) discuss the whole functionality of the slide rule in this article, so matching the most basic information with an likewise basic picture seems beneficial to me. So I prefer the illustration over the photo. I think, for space reasons, we cannot afford two pictures of the s.r. here (actually the previous GA reviewer criticized the article for having too many pictures...). Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   How about discussing "Napier's bones?"

   Undoubtedly a good invention, but how are they related to logarithms? (Other than having the same creator, of course.) For whatever this is worth, Napier's bones points out that they are different.Jakob.scholbach (talk) 22:27, 18 January 2011 (UTC)

   My mistake.Racepacket (talk) 10:45, 19 January 2011 (UTC)

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars, etc.:

6. It is illustrated by images, where possible and appropriate.

   a (images are tagged and non-free images have fair use rationales): b (appropriate use with suitable captions):

   Thank you for contributing the graphs and charts.

7. Overall:

   Pass/Fail:

   I am placing the article on hold. Racepacket (talk) 05:36, 17 January 2011 (UTC)

Please don't overlook the comment in red. Racepacket (talk) 11:09, 19 January 2011 (UTC)

1. Now dealt with (hopefully).

Thanks, Racepacket, for your review! I think all points you raised are now covered. Jakob.scholbach (talk) 21:38, 19 January 2011 (UTC)

Congratulations. Well done. I have corrected the log table discussion, rather than do another round of comments. I hope that you agree. Racepacket (talk) 03:15, 20 January 2011 (UTC)

Kudos to the editors of this now-excellent piece. Way to go! Lfstevens (talk) 00:33, 1 February 2011 (UTC)

Schematic slide rule example

I think the schematic slide rule diagram at the end has been simplified just that little bit too far. I believe this would be much better if it was made of two long rectangles besides each other. This would make it clear that they could be slid relative to each other. Dmcq (talk) 09:41, 2 February 2011 (UTC)

Right. Do you just want to edit/copy that file and add the rectangles (however I don't quite see how the rectangles would make it easier to understand)? First we had a photo of the slide rule where it was difficult to tell what was going on, this is now very schematic. Would a schematic animation be better than all we have so far? If so, I'd volunteer to create one, in the same vein as the current picture. Jakob.scholbach (talk) 21:47, 2 February 2011 (UTC)

I don't think an animation is needed. I was just thinking of clearly showing that the two scales can be slid relative to each other, at the moment it looks like they are part of the same line, two offset rectangles would make the numbers appear more fixed to the rectangles than the line I believe. Dmcq (talk) 23:11, 2 February 2011 (UTC)

OK. Why don't you just edit the image (if this is not used elsewhere, otherwise create a copy)? Be bold ;) Jakob.scholbach (talk) 23:22, 2 February 2011 (UTC)

Fair enough. My illustrations haven't been particularly wonderful but I should be able to manage this so will try tonight if nobody else does first. Dmcq (talk) 14:59, 3 February 2011 (UTC)

I may have beat you to it. Feel free to improve it better. Dicklyon (talk) 07:12, 4 February 2011 (UTC)

At least I hadn't started, I've just come back to do this and I've found you've done it better, I hadn't thought of colouring the two rectangles but that does help considerably I think. Thanks. Dmcq (talk) 11:22, 4 February 2011 (UTC)

Is it possible/desirable to label the distance from 0 to 2 and 0 to 3, respectively, with log 2 and log 3? Same with 6. This way it would be immediately clear why/how logarithms are in this gadget. Maybe also highlighting the 2, 3, and 6 tick would simplify the understanding? Jakob.scholbach (talk) 11:56, 4 February 2011 (UTC)

The image is used in various articles on several wikis, more generally than doing 2 times 3, so if you want to make an annotated one, you'll need to make it a new one, not a replacement. Dicklyon (talk) 05:27, 5 February 2011 (UTC)

Logarithmic derivative

To be comprehensive (an FA criterion), some mention of logarithmic differentiation should be made. Thanks!  Kiefer.Wolfowitz  (Discussion) 00:09, 16 February 2011 (UTC)

It is briefly mentioned in the section on the derivative. Jakob.scholbach (talk) 16:40, 16 February 2011 (UTC)

I missed it in the feast of delicious tidbits. Sorry!  Kiefer.Wolfowitz  (Discussion) 17:17, 16 February 2011 (UTC)

Psychology and functional equations

Would a mathematical analyst review the psychology section, please? A second opinion is needed. Thanks!  Kiefer.Wolfowitz  (Discussion) 17:04, 17 February 2011 (UTC)

My revision avoids the discussion of limits: "Weber's difference equation for the "smallest noticeable change" discretizes the differential equation whose solution is the logarithm function, because the natural logarithm is the integral over dS / S."  Kiefer.Wolfowitz  (Discussion) 14:52, 19 February 2011 (UTC)

Italic e

I believe e should be italic as in the article e (mathematical constant), the article has a DISPLAYTITLE to achieve that, but the quotes have been removed from it in this article, any thoughts? Dmcq (talk) 23:48, 18 February 2011 (UTC)

This has come up here: they were inconsistent but mostly non-italic, in particular the main uses in the lede and section on bases were non-italic, so I changed them for consistency. It's not usual to change formatting to make one article consistent with another, but looking at other articles too they largely seem to use e not e - most of those reached by the template at right for example.--JohnBlackburne^words_deeds 00:11, 19 February 2011 (UTC)

Having a look at that MOS math I also think I disagree with the reasoning about Greek letters. They are not Greek letters when they are in maths, they are maths symbols. Therefore whether they are italic or not should I feel be an agreement in maths not about the Greek language. I think I'll put that argument in the talk page of the MOS and see what people think. I think it is silly to insist that i be upright because it is called iota. Dmcq (talk) 13:42, 19 February 2011 (UTC)

I don't have a strong feeling about the e vs. e. I think in well-typed math papers, constants and functions such as sin, lim, etc (i.e., if they are "constant" in the sense that they have a defined meaning), are not put in italics. Anyway, whoever changes it should do so carefully...

About the Greek letters: MOS is quite explicit. Moreover, LateX does not put Greek letters in italics. Jakob.scholbach (talk) 09:46, 20 February 2011 (UTC)

MOS quite explicitly violates one of the fundamental principles of Wikipedia: verifiability. It states that the majority of mathematical sources do not italicize greek letters? I tried a random selection of mathematical articles from jstor, and it turned out all of them use italic greeks; plus a common sense tells us that the overwhelming number of modern scientific publications are prepared in LaTeX, which by default uses italicized greek font.

On the e / e issue, the policy says that the articles should be consistent in themselves, but not necessarily across Wikipedia. In practice, of course the readers would benefit from having a consistent formatting and style in related articles. The policy was set in place in order to avoid unresolvable disputes or to allow different articles adopt the style that is more common in their field (eg, physicists denote the imaginary unit j; for statisticians X and Y denote known quantities while α and β are the variables; etc). None of these reasons apply here, so we might as well use the italic format, in sync with the main article about e. Another reason is that it is easier to type italic e in LaTeX; and yet another reason is that MoS explicitly recommends using italic e. // stpasha » 10:52, 20 February 2011 (UTC)

I don't care too much about either the e nor the Greek letters. However, any discussions about MOS should be done at the talk there. As long as it stands like this, we should follow MOS (also by italicizing the e, per the link you provided). --Jakob.scholbach (talk) 12:14, 20 February 2011 (UTC)

I have attempted to find all unitalicised e's. Any more should also be italicized. Jakob.scholbach (talk) 19:32, 22 February 2011 (UTC)

What about i’s and d’s — should they be italicized as well (currently we have mixed use)? The same MOS recommends to have them italic too.  // stpasha »  22:05, 22 February 2011 (UTC)

Yes, sure. Could you do it? Jakob.scholbach (talk) 23:12, 22 February 2011 (UTC)

lead image

Arguing that "it doesn't make sense to use a logarithmic spiral as lead image, before showing what a log function is", Dicklyon reverted a recent attempt of mine to make the article more enjoyable for readers who know little about logs (and also reverted a number of uncontroversial edits, which is somewhat unfortunate). To this end, this version featured the logarithmic spiral picture with the nautilus at the top, as opposed to the picture showing the three graphs. My reason for the change was this: the graphs contain too much information condensed in one picture (log_b 1=0, log_b b=1), and also do not tell all that one might wonder about (what about log_b(x) for x negative). On the other hand, the shell is a nice picture that might entice the reader to go on, and it clearly(!) is not necessary to know logs to appreciate such an image. On the contrary, I would say, appreciating and understanding the graphs does require a good understanding of section 1, which some readers won't have (up in the lead). I'm inclined to reinstate my take, any comments? Jakob.scholbach (talk) 07:34, 22 February 2011 (UTC)

I do think that Dicklyon's revert was too hasty, since the edit in question contained many notational improvements, besides just moving the picture. Regarding the picture, both arguments seem valid, so why don't we put both of them in the lead? For the Application section, another picture from logarithmic spiral may be used --- for example the spiral galaxy, or a cyclon.  // stpasha »  09:16, 22 February 2011 (UTC)

I think the graph works better: it shows graphically a fundamental property of logs, that they are are all related by a scale factor. You can also immediately read off notable values such as log(1), log₁₀(10), log(0), and note that it's undefined for logs of negative numbers. Readers new to the subject will find it difficult to interpret all that, but it is all explained later on. You would not expect the graph to show everything but it does show a lot of information in a small space. The nautilus may be a 'nice picture' but it's far less likely readers will get as far as logarithmic spiral and so have it explained, and it has far less mathematical value.--JohnBlackburne^words_deeds 20:47, 22 February 2011 (UTC)

Exactly, we explain it later. In any other situation, we would always put the image next to the text explaining it (in more detail than the image caption). Why don't we do so here? We have the choice of putting the graphs either in the first section (definition) or in the fourth section (log function, analytic properties).

Unexperienced readers (15 y old, say) will not only fail to "interpret" it, they will simply not even remark these features. For example the similarity of the graphs is virtually impossible to guess from the picture, unless you already know it. Similarly, just about 0.1% of all novice readers will remark that the three graphs intersect in one point etc. etc. We would have to explain all of this to make it lucid, but in the lead we arguably don't have the space for this. Right?

About the nautilus: in the article, we never explain the logarithmic spiral (nor should we, given its limited importance). This means, we don't have to wait until the applications section. Yet, the image conveys the idea that logs are meaningful, or good for something. I'm not terribly attached to the nautilus, but we need to have a picture up front which is understandable (a precondition to be meaningful) for the reader jumping into this article. The graphs certainly don't satisfy this criterion. Any suggestion better than the nautilus (and the graph)? Jakob.scholbach (talk) 21:02, 22 February 2011 (UTC)

Citations needed

To finish the remaining obvious obstacles for FA candidacy, we need to find some references (I was the one putting the {{fact}}-tags). Does anyone have access to a good library (especially for older books)? The items still to be covered are this:

• "Related questions in transcendence theory such as linear forms in logarithms are a matter of current research."
• We need a book explaining the method of calculating the log using series, including (ideally) the reduction to small arguments.
• The relation of Napier's logs to modern logs (I remember this to be in Eli Maor's book (already cited), but don't have access to this book right now).
• A qualified statement about the impact of log tables.
• A list of historical log tables. (Goolge reveals this book and this book, but stupidly one cannot browse the content).

Please, anyone who has a library (at home :)), contribute a reference (I volunteer to do the formatting, if you are fed up by this). Thanks, Jakob.scholbach (talk) 20:50, 22 February 2011 (UTC)

Formatting changes

I just undid this change for two reasons. First the manual of style is clear that editors should not change formatting from one style to another without good reason. Second because it made it far more difficult to see the point it is making: it's not a minor example, it's part of the explanation as to how the logarithm works. It was also very poorly formatted with two non-standard fraction formats in one line. The edit summary said "standalone formulas should primarily be used for more important material and avoided if possible for less important" but I've never come across such a rule or guideline.--JohnBlackburne^words_deeds 09:04, 17 February 2011 (UTC)

Hi John, I appreciate your concern about formatting issues, but allow me to explain myself in some more details than allowed by the short edit summary field. First, I do not think that the example with log₂(1/2) is a minor thing, however I believe it bears roughly the same significance as the first example with log(8) and the last example with log(150). By formatting these examples differently we are putting overwhelming (and undue) weight on the second example, since the perceived importance of a fragment of text is proportional to the visibility of that fragment. This is why I say that standalone formulas should be used judgmentally: they serve as an emphasis; and I don't believe the intention of the author was to emphasize the second example at the expense of the first and the third. Also what's so non-standard about the template {{frac2}}? In any case, it might be better to write the expression as log₂ 0.5. // stpasha » 19:02, 17 February 2011 (UTC)

They do not serve as emphasis. The main reason a formula is rendered using LaTeX is usually complexity or size: an integral or a fraction for example. Another reason is consistency, so all similar formulae look the same, which applies here as it's consistent with the formula above and the examples below. It's position in the article and the text around it and the text around it determine its importance, and this is an useful example which is worth laying out clearly. You are right about the standard (I was thinking of {{frac}}), but even if standards compliant it was ugly with varying baselines, two fraction styles, and wide spacing between the lines. And an overriding principle in articles is stability. You should not change from one format to another simply because of your judgement. Nor do I see how you know the "intention of the author" is for something different.

We've already had this discussion, or a related one on HTML vs TeX, which resulted in the current formatting. See #Formula formatting consistency - informal RFC above. This was part of the process of getting it ready for GA, which it passed. Given the consensus that was arrived at for the current formatting, and the acknowledgement of it's quality with the GA award, it should not be changed back to HTML formatting without consensus being achieved for the change first.--JohnBlackburne^words_deeds 19:53, 17 February 2011 (UTC)

Even though one of the reasons a formula may be rendered using LaTeX is complexity / size, still doing so puts an emphasis on that formula. Simply because the formula stands on its own line, is surrounded by large amounts of whitespace, and is rendered in a bigger fontsize. Sometimes we use this effect deliberately (eg, first displayed formula in that section --- it is simple enough to generate in HTML and put in-line, but as it is the definition of the logarithm we want it to be much more noticeable), sometimes it may be used to break the monotonicity of what otherwise would be a wall of text (eg, the Music section), but sometimes the use of unwarranted displayed formulas is just bad style. As far as I can see this happens twice within the article, second time is with the $\scriptstyle {\sqrt {-5+{\sqrt[{3}]{3/13}}}}$ number (btw, was it intentional to have a negative number under the root?)

I agree that varying baseline looks rather ugly. As an alternative we can write log₂0.5 = log₂(1/2) = log₂(2⁻¹), which avoids having two fractions with varying baselines. Anyways I understand that you might feel frustrated over another argument regarding formatting issues, however it is a different topic from the one discussed above: I argue about the choice between inline and standalone formulas, whereas the previous discussion was about which formatting to use for each of those formats. // stpasha » 21:13, 18 February 2011 (UTC)

It seems your come down to your opinion, which is never enough reason to change the formatting as described at WP:STABILITY. Apart from that although no other editor has replied the editor immediately after expressed Agreement with the previous edit, so we already have a third opinion on this.--JohnBlackburne^words_deeds 23:17, 18 February 2011 (UTC)

You can add me to the ones supporting the current format. I don't think the change to using the frac2 template and html looked as good. 23:37, 18 February 2011 (UTC)

Ok then, how about this suggestion:

For example, log₂(8) = 3 (pronounced as "the logarithm of 8 to base 2"), since 2³ = 2 × 2 × 2 = 8. Logarithms of numbers less than one are negative: log₂(0.5) = −1, since 2⁻¹ = 1/2 = 0.5 (the first equality is because a⁻¹ = 1/a is the reciprocal of a ^{[nb 1]}). If a number is not an exact power, its logarithm will be non-whole: log₁₀(150) ≈ 2.176. The value of the logarithm lies between two and three, just as 150 lies between 10² = 100 and 10³ = 1000.

// stpasha » 00:52, 19 February 2011 (UTC)

I am planning to overhaul the first section, and to include a short section on how exponentials work. This will simplify the article for readers unaware of that concept. The above formatting discussions are probably premature at this point; but I also prefer the current version. Jakob.scholbach (talk) 09:40, 20 February 2011 (UTC)

You mean explain that 2³ is 8? I think that is a bit over the top and moreover it is already explained in the first sentence in the lead complete with a link to the exponentiation article. Dmcq (talk) 12:16, 20 February 2011 (UTC)

I want to add a short opening paragraph, and want to move the footnote with the remark about a^-1 etc. into the proper text. The explanation about 2^3 is mostly to establish/recall the notation of an exponent. I believe, when we can, we should not rely too much on wikilinks. --Jakob.scholbach (talk) 12:20, 20 February 2011 (UTC)

and again

I just undid this change as I thought we had established a consensus for the current formatting, i.e. for LaTeX formulae not inline math. And again I would point out that changing from one allowed style to another, with no good reason other than personal preference, is never acceptable according to the manual of style or the associated arbitration motions.--JohnBlackburne^words_deeds 02:11, 22 February 2011 (UTC)

John, you are being counterproductive. Have you even looked at the edit before you reverted it? I'm organizing content, adding additional examples, adding explanations and comments to the examples; and then I alter formatting because with the new material, old formatting doesn't look so good. But if you insist so much on LaTeX formulas, I'll make them LaTeX. // stpasha » 02:26, 22 February 2011 (UTC)

Please play nicely!!!!! (In a hundred years, we'll be dead, so notation isn't worth a fight.)  Kiefer.Wolfowitz  (Discussion) 02:40, 22 February 2011 (UTC)

I also have mixed feelings about the MOS-compliance of the edit of stpasha in question: please use prose whereever possible, and do not highlight things in bold face. Jakob.scholbach (talk) 07:23, 22 February 2011 (UTC)

I agree, boldface was probably a bad idea, and 1/2¹ = 1/2 was another bad idea, but please don't throw the babies with the bath water, can you imagine how discouraging it is? // stpasha » 08:31, 22 February 2011 (UTC)

Again, we have three editors, me, Dmcq and Jakob.scolbach who have either objected to removing the <math> examples or expressed a clear preference for them. As that seems to me a clear consensus, compared to the one editor who keeps removing them, can they now be put back in and left in. What's there now is a dense and unattractive mess compared to what was there before.--JohnBlackburne^words_deeds 20:29, 22 February 2011 (UTC)

Since the beginning of this argument we went through a number of choices for the presentation of examples, and the current choice is to show them as a list. This might look condensed, but in my opinion it is better than the prose because this way the beginning and the end of each example is clearly visible. A person reading this section may want to skip some of the examples because they are too easy, or too hard, or he's looking for some particular example (like log(1)). With a list it is easy to do, with prose much harder.

The reasons the old version (which you keep reverting to) seems unacceptable to me is because different examples within the section were formatted inconsistently; and also because with the addition of few more examples the section would be hard to read (unless you believe that the new examples should not even be there as they do not aid understanding the material).

Please, John, be civil. I do not understand why you insist on edit-warring. If you think the current version is not good, then go ahead and improve it. But do not delete it simply because you don't like it. // stpasha » 21:41, 22 February 2011 (UTC)

I'm not sure why you think I'm uncivil. I accept you are trying to improve the article, I just think what you've done has made it worse as I've described. I'm not sure why you think I'm edit warring: you've removed the <math> formatted examples three times, clearly against consensus as indicated by the diffs above. But I don't like repeatedly reverting and have not done so this time, in the hope that other editors can help resolve this.--JohnBlackburne^words_deeds 22:07, 22 February 2011 (UTC)

I also don't think anyone is uncivil in this matter. I do think, though, that artificial spaces (log_b(b) = 1) look ugly, as do the fractions (in text mode). Moreover, the guideline WP:EMBED cited by stpasha's edit summary does not seem to back up your edit (it says "While prose in general is preferred for the writing of articles, there are occasions when some form of list may be appropriate", the only thing which remotely resembles our situation would be the section "Children" of WP:EMBED, but I do prefer the prose in this place. The difference between the (unbulleted) list of examples in the next section is that there we don't have any accompanying prose). Moreover, "If a number is not an exact power" is unprecise (and would be difficult to make precise at this stage). Finally, we have to be careful with WP:NOTTEXTBOOK, so should not give too many examples (log0.5(2) = −1 is one too much, I believe). Finally, the prose is weak ("when the base b > 1."). All in all, I did not consider stpasha's (repeated identical) edit in this particular section to be an improvement, which is why I merged (not deleted!) his content with the previous variant we had. As far as I can see, stpasha, you are the only one consistently reverting to your version. Any comments? Jakob.scholbach (talk) 23:11, 22 February 2011 (UTC)

Yes, WP:EMBED is the relevant policy, i.e. it describes more clearly when lists in non-list articles are appropriate. What we have now is a section of prose roughly chopped up into a list by bullet points, which alone is an inappropriate use of a list and makes the prose much more difficult to follow. The inline math makes for a very poorly formatted list with in the embedded fractions and superscripts breaking the line widths. And there's some very unusual HTML in there which is not normally used with formulae and will only make them harder to edit, as per MOS:#Keep_markup_simple.--JohnBlackburne^words_deeds 23:39, 22 February 2011 (UTC)

Sorry I haven't been following this closely enough, but I did a revert of what looked like an inappropriate style change, based largely on a disconnect between what I saw and what the edit summary said. I'd just like to say that we need to keep it simple. No "spans" and such html stuff where wiki markup will do, even if it's a bit ugly. And use LaTeX for display math, but avoid it inline when possible. Dicklyon (talk) 00:29, 23 February 2011 (UTC)

In the browser I'm using at work, the fractions in the markups used by stpasha look really ugly (I was unaware of that earlier): the numerator is shown where usually the denominator is placed, and then under the numerator we have the - and the denominator. Really, we cannot use text markup for fractions. Jakob.scholbach (talk) 12:40, 23 February 2011 (UTC)

I went back to the original version, for the reasons mentioned above. I hope we can all (including stpasha) live on with that. Jakob.scholbach (talk) 19:12, 23 February 2011 (UTC)

Michael Stifel

The "predecessors" section mentions Michael Stifel's work and provides 2 references. The first of them ([70] as of right now) says the following: "It used to be said that Stifel was real inventor of logarithms, but it is now certain that this opinion was due to a misapprehension of a passage in which he compares geometrical and arithmetical progressions". The second reference ([71]) appraises Stifel's work, and as an example gives his "table of logarithms", which looks to me exactly like a comparison between arithmetical and geometrical progressions. In view of this, I think Stifel should not be ascribed as the inventor of logarithms, but perhaps as a person who used to be mistakenly believed to be an inventor. // stpasha »  21:49, 22 February 2011 (UTC)

Interesting. I have crawled a bit more: all other sources I have seen so far do acknowledge Stifel's thing as a rudimentary log table EOM (scroll down a bit), this book, this book (written by Felix Klein). Given that there is only one source claiming the contrary, and given that it is quite old (authored in 1908), I think "that has been considered" is a fair summary of the situation, as far as I can tell.

I will replace the only reference claiming the contrary by the EOM entry. Jakob.scholbach (talk) 19:24, 23 February 2011 (UTC)

ready for FA??

I'm planning to nominate the article for FAC in the next few days. Are there any objections to this? If not, would somebody kindly help out at the FAC process (usually there are tons of nitpicky editing requests...) Jakob.scholbach (talk) 22:21, 16 March 2011 (UTC)

I'm keeping an eye on it and have watch listed the discussion page, so if anything comes up I think I can help with I'll step in.--JohnBlackburne^words_deeds 22:40, 21 March 2011 (UTC)

Thanks for dealing with all the nit-picky requests; I hope you don't mind that I jump in with my own tweaks from time to time. Dicklyon (talk) 04:10, 25 March 2011 (UTC)

Purpose?

I didn't understand the statement "The purpose of logarithms is to undo the operation of raising a fixed number to a certain power." This is not a purpose. I did a book search, and the only "purpose of logarithms" I could find was to facilitate calculations. I realize that's not much of a current purpose, so I added it as the "original purpose", and changed the sentence above the "function." This may not be the best fix, but it's better, I think. Dicklyon (talk) 04:08, 25 March 2011 (UTC)

New plot

improvement?

I tried to make a better plot, with less confusing colors, and more normal bases. But Matlab's plot2svg mangled some of the text, and seems to have other issues. I can get a good .eps file out, if someone knows how to get a good .svg from there. Anyone want to switch to this? Dicklyon (talk) 06:35, 9 February 2011 (UTC)

Yes, that's good. Thanks for creating it. Is it possible, though, to rotate the caption (maybe also with a bigger font) of the y-axis and place it, say, in the huge white space under the x-axis? Also, the subscript b should really be a subscript. AFAIK the file does not have to be an svg. Jakob.scholbach (talk) 08:58, 9 February 2011 (UTC)

P.S. Or just label the three graphs with log₂(x) etc.? Jakob.scholbach (talk) 08:59, 9 February 2011 (UTC)

As for colorblindness questions: KSmrq has at his page a list of colors that are (supposedly) optimal to distinguish. You could consider using these instead of the standard R, G, B. Jakob.scholbach (talk) 09:02, 9 February 2011 (UTC)

Like one of my uncles when given a new pullover 'Thanks very much, red and green are my favourite colours', of course he was asked which one that was then :) I'll try and remember that page even if I don't think it matters too much here because they are continuous smooth lines. Dmcq (talk) 11:23, 9 February 2011 (UTC)

I think I can make it work by putting the log₂(x) etc. on the curves (matlab correctly renders the subscript in going to .eps, but not to .svg; I think I can convert that to .svg if I don't have rotated text or try to edit it, but not sure yet). The colors become irrelevant if the curves are labelled, but I'll try to make them distinguishable. Dicklyon (talk) 19:07, 9 February 2011 (UTC)

What is the problem, just to make a subscript? It is quite easy, upload your SVG and I will fix text inscriptions. Incnis Mrsi (talk) 00:26, 16 February 2011 (UTC)

PNG version with better text

Can't get the svg to behave. Try this PNG one? Dicklyon (talk) 19:44, 9 February 2011 (UTC)

Great, thx. Be sure to update the image caption in the article accordingly. Jakob.scholbach (talk) 21:31, 9 February 2011 (UTC)

Done. I predict that next it will be tagged as needing an SVG replacment, and then it will be replaced by a less-good SVG version, whether mine or the old one. If someone knows how to make good SVGs they might be able to head this off. Dicklyon (talk) 02:16, 10 February 2011 (UTC)

The SVG that is was replaced with is actually quite nice (the one now showing above). But I question the placement of the new dot: why not at (0.5, 1) to show the same relationship as the others? Dicklyon (talk) 18:50, 25 March 2011 (UTC)

I actually prefer the previous example, showing only bases 2, e, and 10. Logs for bases < 1 are hardly ever used, it seems. So they are more of "scholarly" interest. Also, the article does make clear later on that log's with such bases are strictly decreasing. Finally, if we have to take this example, it should certainly feature the dot at (.5, 1), not at (2, -1), just to make it clear the analogy with the other bases. Jakob.scholbach (talk) 19:00, 25 March 2011 (UTC)

"on a scale of 1 to 10"

This is misleading. Theoretically, earthquakes can measure less than 1, or more than 10, on the Richter scale. 206.225.134.57 (talk) 12:02, 27 March 2011 (UTC)

I think this minor imprecision (which is, as you point out, only of theoretical importance anyway) does not harm us here. After all, the lead has to summarize the aritcle in an accessible way, and the phrase captures very well, IMO, the ability of logs of boiling down large scales to smaller ones. Jakob.scholbach (talk) 13:05, 27 March 2011 (UTC)

The limits are somewhat arbitrary, but not unreasinable. An event of magnitude less than 1 is never called an earthquake, I think, and there seem to be theoretical reasons why we'll never see an earthquake as large as 10. Dicklyon (talk) 17:19, 27 March 2011 (UTC)

FAC comments

-Lead

• In lead picture, don't use serial comma for consistency
• Be consistent with "to the base X" and "to base X"
• I find "is written" neater than "is written as" (and in rest of article).
• The sentence "The logarithm of a product of two numbers equals the sum of their logarithms" comes out of the blue. Maybe introduce with "A fundamental property is ..."
• Richter scale problem (see above). Another problem is that the Richter scale doesn't "measure earthquakes".
• "The natural logarithm uses the constant e (approximately 2.718) as its base, and is especially widespread in calculus." I would also mention that the natural log is the one primarily used in (pure) mathematics.
• The fact that the logarithm is a mathematical function (yet even a mathematical object!) is hidden under the rug. This is too much of a dumbing-down.
• "primarily aids computing applications" -> The untrained eye might find this very confusing, interpreting "computing" as the verb "to compute". Maybe rephrase as "is used in computer science". Also, I don't think that logs base 2 are primarily to "aid computation". They are simply the most natural in the field.
• "Logarithms are commonplace in scientific formulas, measure the complexity of algorithms and of fractals, and appear in formulas counting prime numbers." Remove serial comma for consistency.
• "They describe musical intervals, inform some models in psychophysics and can aid in forensic accounting." This seems to be an arbitrary mash-up of examples, which is to me original research.
• "The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers." By saying "generalizes the logarithm" here, you are implying that "logarithm" here is defined as the real logarithm. This should be made clear right from the definition, by changing "The logarithm of a number..." to "The logarithm of a real number..."

-Later

• "Roughly, a differentiable function is one whose graph has no sharp "corners"." This is too much dumbing down (or plain wrong!) to my taste. The sign function is not differentiable, yet really doesn't have "sharp corners". Randomblue (talk) 00:45, 28 March 2011 (UTC)

In the interest of a single discussion about FAC, I pasted your comments to the FAC page and responded there. Jakob.scholbach (talk) 16:15, 28 March 2011 (UTC)

trivia

In case relevant:
these are probably inconsequential for article, but thought no harm in posting.

"logarithmic operators" are used in numerical analysis, theoretical physics and computer graphics, with various meanings. See, e.g. [2] and:

Extended content

Inversion of a logarithmic operator defined on a regular set of arcs lying on a circle Author(s): Il'inskii AS, Chernokozhin COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Volume: 49 Issue: 8 Pages: 1415-1428 Published: AUG 2009 Singletons and logarithmic CFT in ADS CFT correspondence Author(s): Kogan II Source: PHYSICS LETTERS B Volume: 458 Issue: 1 Pages: 66-72 Published: JUL 1 1999 Times Cited: 36 The Haldane-Rezayi quantum Hall state and conformal field theory Author(s): Gurarie V, Flohr M, Nayak C Source: NUCLEAR PHYSICS B Volume: 498 Issue: 3 Pages: 513-538 Published: AUG 11 1997 Times Cited: 63 Title: A FAMILY OF NEWTON TYPE ITERATIVE PROCESSES Author(s): HERNANDEZ MA, SALANOVA MA Source: INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS Volume: 51 Issue: 3-4 Pages: 205-214 Published: 1994

Formulas for integrals containing logarithms are listed e.g. in the Gradshteyn and Ryzhik compendium Michael P. Barnett (talk) 17:13, 3 May 2011 (UTC)

Two reversals of exponentiation

Exponentiation (involution) is noncommutative, so it has two reversals: logarithms and roots (evolution).
—Wavelength (talk) 15:23, 10 May 2011 (UTC)

Involutes and evolutes mean something different, perhaps you meant inverse? Anyway wahat was the point of what you said? Dmcq (talk) 16:51, 10 May 2011 (UTC)

The words involution and evolution in my first message can be disregarded. (Exponentiation#History of the notation says: "Another historical synonym, involution, is now rare and should not be confused with its more common meaning.")

"Logarithm" mentions twice that logarithms reverse exponentiation, but the article does not mention that roots also reverse exponentiation although in a different way.

—Wavelength (talk) 18:32, 10 May 2011 (UTC)

The article is about logarithms, not exponentiation. Dmcq (talk) 22:58, 10 May 2011 (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 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 complex series approximation in that most ordinary people are not going to know anything about it, yet the complex series approximation 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 logarithms to possess. What are your thoughts/concerns about adding this section?Maonaqua (talk) 14:16, 12 May 2011 (UTC)

See Talk:Natural logarithm#The J-ln and Eulers triangle - best to keep discussion in one place. Gandalf61 (talk) 14:27, 12 May 2011 (UTC)

First sentence

I confess I didn't read every word of the debates at the FAC over the first sentence or two, but I think the current version has some flaws. As I write this, the current version is "The logarithm is the exponent by which a number called base has to be raised to produce another number." A couple of things that bother me: "exponent by which" is a little surprising; it's much more usual to see "power to which". I also don't like "a number called base"; it feels like it should be "the base" or "a base". Finally, "another number" is odd; it has no prior referent in the sentence so it takes a second or two to work out what's meant.

Here's a definition from an 1875 textbook: "A logarithm of a number is the exponent denoting the power to which a fixed number, called the base, must be raised in order to produce the given number". I like several things about this. The phrase "the exponent denoting the power" gets around my first comment above. Putting "called the base" in parenthetical commas allows the main course of the explanation to proceed without the new term "base" as part of the explanation. Finally, "of a number" at the start gives us a referent for "the given number" at the end, which feels more coherent.

I've supported at FAC since I think this is a minor point, but I do think the first sentence can be improved. Thanks for a fine mathematics article, by the way; these are underrepresented at WP:FA and are important articles for the encyclopedia. Mike Christie (talk - contribs - library) 13:55, 28 May 2011 (UTC)

I've incoporated most of your suggestions. The only thing I did not take up is the exponent vs. power thing: in the expression b^x, x is the exponent, b^x is the x-th power of b (or b raised to the x-th power). A number of people seem to be confused or sloppy about this, including myself at some point. However, this is the correct usage of the terminology. Secondly, I find "exponent denoting the power" difficult to understand. "Denoting" is used in a sense that I consider less standard than, say, "produce" (which is what we currently have). Jakob.scholbach (talk) 15:30, 28 May 2011 (UTC)

That does look better; thanks. I see the distinction you are making with regard to "power" and "exponent", but isn't it more usual to say "the exponent to which a number is raised than "by which"? I'm OK with leaving it as it is if you are confident that it's standard usage, but it looks odd to me. Mike Christie (talk - contribs - library) 16:13, 28 May 2011 (UTC)

After some discussion at the FAC page, I got convinced that the most commonly used wording is "raise a number by an exponent" or "raise a number to a power". So, "the exponent, by which " is the right, I think. Jakob.scholbach (talk) 17:12, 28 May 2011 (UTC)

Citation needed?

A recent edit of Geometry guy added a "citation needed" to the sentence "logarithms can be calculated using power series or the arithmetic-geometric mean or retrieved from a precalculated table that provides a fixed precision.". I don't understand this request: power series, AGM are mentioned just below (and referenced), while the tables are mentioned above (and referenced). What specifically do you think needs a reference? Jakob.scholbach (talk) 05:40, 30 May 2011 (UTC)

Also, Zhang, M.; Delgado-Frias, J.G.; Vassiliadis, S. (1994), "Table driven Newton scheme for high precision logarithm generation" (currently ref no. 41) is right next to the tag, and does mention all these methods. Jakob.scholbach (talk) 21:26, 30 May 2011 (UTC)

A general reference to a reliable secondary source would be helpful at the beginning of a section like this. The citation at the end of the paragraph is not "next to" the tag, is a primary source, and does not mention the AGM approach. Geometry guy 22:10, 30 May 2011 (UTC)

OK. Two refs given. Jakob.scholbach (talk) 17:22, 31 May 2011 (UTC)

"Factor" in the lead

"For example, the logarithm of 1000 to base 10 is 3, because three factors of 10 must be multiplied to yield a thousand"

I don't believe factor is being used here correctly: for elementary mathematics, "factor" is usually limited to prime factorization or divisors. I'm hardly halfway capable with mathematics though, so I wanted to defer the question to the primary author(s): Should the sentence be reworded? NW (Talk) 16:21, 25 March 2011 (UTC)

Factors are not necessarily prime factors. Dicklyon (talk) 16:35, 25 March 2011 (UTC)

Sure. But the average reader probably wouldn't equate "three factors of 10" and "{10, 10, 10}". Is there a way this could be reworded for the uninformed like myself? NW (Talk) 17:24, 25 March 2011 (UTC)

I'm not a native speaker, so cannot really judge. But multiplication, for example, says: The numbers to be multiplied are generally called the "factors". What would you call them instead? Jakob.scholbach (talk) 17:51, 25 March 2011 (UTC)

Perhaps he's worried about the "of", making him think about 1, 2 and 5. Would writing three factors "10" be better? −Woodstone (talk) 03:21, 26 March 2011 (UTC)

Right. three factors "10" sounds OK; would the follow phrase work instead: For example, the logarithm of 1000 to base 10 is 3, because the product of three copies of 10 is 1000? I'm not terribly sure if that is the case, but three factors "10" sounds gramatically incorrect for some reason. NW (Talk) 04:08, 26 March 2011 (UTC)

NW, I think you are mistaken in your conception that common people would associated factors with prime factors. Aside from the fact that most common people probably wouldn't even know (exactly) about primes the term "factor" is frequently used in the media for statements like "Radition increased by a factor of 1000" (say). Both examples given above are not improvements, IMO. Nageh (talk) 07:55, 26 March 2011 (UTC)

I can see that "factor" is not very clear in this context. Is "instances" better? Nageh (talk) 12:32, 27 March 2011 (UTC)

I don't think "factor" is a problem. Only people knowledgeable about prime factorization and the like will even think of the factors 2 and 5 of 10. This readership will already know what logs are about, so we can safely commit this minor ambiguity of notation at this point without disturbing them. Secondly, right in the next sentence it is explained what is meant by the three "factors". I think this is as clear as can be. "Instance" is, to me, a non-established term, and I don't think there is a reason to deviate from standard terminology here. Jakob.scholbach (talk) 13:02, 27 March 2011 (UTC)

I agree. Tijfo098 (talk) 06:05, 17 April 2011 (UTC)

Why not just use "because three 10s must be multiplied"? — Preceding unsigned comment added by 71.227.176.141 (talk) 19:59, 5 June 2011 (UTC)

The Decibel

The introduction states

"the decibel is a logarithmic unit quantifying sound pressure and voltage ratios"

Whilst this is true, it is not the full story. The decibel quantifies power ratios, and only finds applications in the above. — Preceding unsigned comment added by 86.185.154.30 (talk) 08:40, 5 June 2011 (UTC)

In the interest of a short and concise lead (see WP:LEAD), we cannot mention everything here. The power ratios are mentioned in the text below. Jakob.scholbach (talk) 16:28, 5 June 2011 (UTC)

... and psychophysics ...

I think we should not call Stevens' Power Law 'more precise' than anything. It has long ago been demonstrated to be false. It isn't that 'recent' either, developed in the fifties and sixties. Stevens was a bit of a fraud. If you don't believe me, read Poulton (1989). I would cut any reference to Stevens' work here.

Reference: Poulton E. C. (1989) Bias in Quantifying Judgements. Hove & London: Erlbaum

Vronks (talk) 09:20, 5 June 2011 (UTC)

According to a number of citations, including the ones we give, Stevens law is deemed more precise than Weber-Fechner. There may be other, even more accurate laws (depending on the particular situation etc.), but this is nothing this article should talk about. Jakob.scholbach (talk) 16:29, 5 June 2011 (UTC)

Napier's contribution ?

I wish the historical section would clarify the advances that Napier brought. He seemed to have only calculated numbers for 20 years: instead the article should clarify the method he proposed. Did he show that the L he calculated could simplify calculations ? How ? Carbo1200 (talk) 11:47, 5 June 2011 (UTC)

The article says "The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention." This, i.e., the tables, is what enabled the practical use of logs. None of the sources I've met mentions Napier creating such tables. Insofar I think the presentation is OK. Jakob.scholbach (talk) 16:34, 5 June 2011 (UTC)

Since -> Because

Is there some standard that says since must be changed to because everywhere, because that's what's happened in the article. I use both words myself and only use because for emphasis. Dmcq (talk) 13:28, 5 June 2011 (UTC)

I was wondering the same. I also consider changing "is due to" to "comes from" not an improvement. But then I'm not a native English speaker. Nageh (talk) 15:07, 5 June 2011 (UTC)

In formal English, "since" just describes a time order with a past event. John is happier since the year began. (Beginning of the year may have had nothing to do with John being happier; it's just a time marker.) "Because" asserts a cause (a causal relationship). John is happier because he married. (The marriage caused John to be happier.) That distinction is often lost if the "since" event in the past is also the cause. John is happier because/since he married. In many informal uses, "since" and "because" have blurred into near synonyms. Under stricter rules, "since" is often misused where "because" is more appropriate.

Changing "is due to" and "comes from" can be viewed as reducing wordiness. But see random usage comments.

Glrx (talk) 16:25, 5 June 2011 (UTC)

I believe this is incorrect; "since" has more than one meaning, and it can be an exact synonym of because. I think it would be a mistake to universally change "since" to "because" where that's the intended meaning. Mike Christie (talk - contribs - library) 16:34, 5 June 2011 (UTC)

Thank you for the information. Though the very link that you provide states that "since" and "because" can be used pretty synonymously these days. Nageh (talk) 18:02, 5 June 2011 (UTC)

The OED says you are wrong. Since be used to mean "Because" and has been used in that way, since at least the 16th century.TR 10:17, 6 June 2011 (UTC)

Comment: in English-language writing about mathematics, it's common to use multiple synonyms so that the prose style doesn't become too repetitive. As well as since/because, we have therefore/hence/thus/so, and it's not hard to come up with other examples. If there are two words that mean the same then it's a good idea to use both on occasion, as long as it doesn't cause confusion. In this case I think there's no danger of confusion. Jowa fan (talk) 10:33, 6 June 2011 (UTC)

I'm not advocating anything. The question was about "some standard". I explained why some speakers would view some uses of since as informal.

A dictionary is not a style guide. In the dictionary that I consulted, since meaning because is the third definition -- after an obsolete definition.[3]

A search shows that distinctions are present.

...

'Since' means the same as because. 'Since' tends to be used in more informal spoken English. Important note: "Since" when used as a conjunction is typically used to refer to a period of time, while "because" implies a cause or reason.[4]

...

What is the difference between since and because?

'since' can also be used to mean 'for the reason that' or 'because' as well as referring to time. This can cause confusion unless the writer is careful:

"Since Paul left university last year, he has no academic qualification."

When you start to read this sentence, it seems to mean "Ever since/from the time he left last year,..." but it really means "Because he dropped out of university, he has no degree."

I reached the office earlier because there was less traffic.

I was unable to log into the website because the server crashed.

Because of the risk of confusion, and at this stage of learning English, it might be better to use 'because', and use 'since' only when referring to a period of time.[5]

...

The NOAD reports a note about the usage of since.

When using since as a causal conjunction to mean because or given that, be aware that in some contexts or constructions the word may be construed as referring to time. For example, in the sentence "since Mrs. Jefferson moved to Baltimore in the 1990s, she was not aware of the underlying complexities", it is not clear, especially at the beginning, whether since means "because" or "from the time when". It is often better to simply say "because", if that is the intended meaning.[6]

Glrx (talk) 15:28, 7 June 2011 (UTC)

Thanks, I think that was clarifying. So "since" and "because" can be used synonymously unless it may create the sort confusion described above. Nageh (talk) 17:12, 7 June 2011 (UTC)

Which is probably why "since" is almost never seen as a problem in math writing, which is usually a sequence of logical deductions outside of time. I, for one, would like to see the "since"s returned to this article. RobHar (talk) 17:27, 7 June 2011 (UTC)

I use because where something special has to be noticed rather than straightforward workings. Anything straightforward I'd just say since or from. Dmcq (talk) 09:02, 10 June 2011 (UTC)

Copyedit required

In the sentence "The base of the natural logarithm is the constant e (≈ 2.718). It is widespread in pure mathematics, especially calculus." it reads as if 'it' refers to 'the base'. 81.107.33.85 (talk) 15:51, 5 June 2011 (UTC)

Good point. Now reworded. Jakob.scholbach (talk) 16:36, 5 June 2011 (UTC)

Introduced by Al-Khawarizmi?

I just deleted the sentence Logarithms were introduced by the great Muslim mathematician 'Abu Muhammad Musa Al-Khawarizmi'. (ETA: actually I didn't delete it, I pressed "preview" instead of "save". But someone else has deleted it now.) I guess this refers to Muḥammad ibn Mūsā al-Khwārizmī. I checked that page and couldn't find any mention of logarithms. If Al-Khawarizmi did in fact invent/discover logarithms, it should be mentioned there; we can add this fact to the logarithm page once a suitable reference for it has been found. Jowa fan (talk) 05:02, 10 June 2011 (UTC)

As far as I know, Al-Khawarizmi is credited for the notion of algorithms, which has, except for the phonetic similarity, nothing to do with logarithms. Jakob.scholbach (talk) 05:49, 10 June 2011 (UTC)

I think he's actually credited with transmitting the notion of algorism (calculation with decimal numbers) from India to Europe by way of the middle east. The word "algorithm" comes from his name via that word. According to the Online Etymology Dictionary [7], the etymology of logarithm is entirely unrelated, being a portmanteau of logos (meaning in this context ratio) and arithmos (number). —David Eppstein (talk) 06:07, 10 June 2011 (UTC)