Wikipedia:Peer review/Logarithm/archive1

Logarithm

This peer review discussion has been closed.
I've listed this article for peer review because this article recently passed its Good Article Nomination and I'm planning to push it to FA level. Any comments welcome.

Thanks, Jakob.scholbach (talk) 21:38, 9 February 2011 (UTC)

Can I suggest toning down the first sentence a little bit? Perhaps by recasting the logarithm as a relationship before fleshing out some technical details, for example:

The logarithm is a relationship between the numbers a,b, and c in the equation a^b = c. The relationship is defined as follows: given a^b = c, the logarithm of the number c, with respect to the base a, is the exponent b. For example, since 4² = 2⁴ = 16, the logarithm of 16 with respect to the base 4 is 2, and with respect to the base 2 is 4. This relationship is denoted log_ac = b and (enunciated?) the log of c to the base a is b.

This proposed sentence is terrible. If the logarithm is a "relationship", then that "relationship" is already expressed by a^b = c. Michael Hardy (talk) 04:48, 14 February 2011 (UTC)

I expect most people looking this article up will have had limited expose to reading mathematics. An afterthought: maybe my examples are confusing since most of the numbers are the same. I'll try and add more comments later. Cheers, Ben (talk) 11:47, 10 February 2011 (UTC)

Thanks for your suggestion; however I'm not sure the detour (and disguised link) to a function/relation is helpful and/or necessary. The logarithm is, most basically, really just a number. Isn't it? Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

Two suggestions about the definition of the logarithm

(Which are not contradictory with the previous suggestion, as far as it could be suggested to bring this formal sentence further down in the article, and rewrite it in a less formal style in the lead).

My comments would be about the question "but what is precisely the logarithm ?" that a beginner could be expected to be interested in, and for which an answer could reasonably be wished in every maths article, be it a highschool topic or a research topic.

First, the definition is not obvious to find - as far as I looked for it, I finally found it in the first sentence of the lead, but nowhere clearly in the article proper. This is not very suitable, such an important thing could be retold in the article proper - or even sent in the article and replaced in the lead by something a bit more informal, as suggested by Ben above.

Thanks, someone has apparently removed it from the first section. I'll reinstate it there at once. Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

Then, supposing I am a highschool student looking for precise information, and supposing I have understood the definition lies in this first sentence, I don't fully understand it. I click on the wikilink on base. Oooops, this brings me to an article titled radix, which seems in casual reading to be about "bases" in elementary arithmetic - if I read it thoroughly, I discover relevant information in the bottom, but it was not obvious where to find it. If I click on power, I am redirected on exponentiation and here again I might be a bit lost - am I supposed to understand by my own that the place to read is the subsection called "Real powers" ? The two first wikilinks are not informative enough (I have not tried the following ones). As I may seem a bit negative, here follows a suggestion (I practise it as often as I can when I edit maths articles, mostly on :fr wikipedia) : use redirects towards sections. In the article radix a section called "Radix in analysis" can be created (even should, since the information is badly classified under the title "In Arithmetic". Then a redirect called [[radix (analysis)]] (or something similar) sending to [[radix#Radix in analysis]] may be created, and at last the link in the logarithm article can be reinstated more precisely as [[radix in analysis|base]]. The same can be done with a redirection to section which may be called [[power with real exponent]], sending on the relevant section of the big exponentiation article.

I removed the wikilink to base/radix (which was superfluous there) and changed the "power" to "exponent". (The suggestion with the subsection might work, but ideally we would not need such a technique.) Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

More generally, a complete review of every wikilink in the article should be underdone before label is asked. Circular references are the plagues of maths articles, and I would not be suprised that a few ones haunt this place ; even if they don't, making wikilinks as precise as possible is an important aim.

OK, I'll do that. Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

Now another topic about the definition : NPoV. The present definition is sourced by only one book, probably well suited to this purpose, Basics Of Mathematics by Kate, S.K.; Bhapkar, H.R. As an experienced reader, I know that for other sources, the logarithm is the primitive of x $\mapsto$ 1/x that vanishes at 1. (I suppose this is the choice of a minority of sources, but certainly a significant minority). As long as we consider these "points of view" as different points of view about the logarithm, they are both adressed in the article, which respects NPoV. But if we consider that they are points of view about the definition of the logarithm, NPoV is not respected. Due to the expected mastery of mathematics of some readers of such an article, I quite think we are in the second branch of this alternative.

Did you see that the article characterizes/redefines logarithms as the integral in section 4.3? If yes, I'm not quite sure I can follow you about the NPOV. The first definition we give is clearly not POV, it is just the usual definition any elementary math book contains. Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

Yes, I did of course, and I underlined I knew it was already here, but not already here pointed as a possible definition. And as I explained in the following too longish paragraphs, I understand very well there are good reasons to answer me "no sorry". But I don't agree with your "it is just the usual definition any elementary math book contains" -> it might be a difference of culture between undergraduate level teaching in the US and in France, but here (in France) it is fairly common to begin an exposition by building the logarithm as a primitive, then the exponential as a reciprocal, and at last the powers from the formula a^x=e^x ln a. Here is an example available on the web : [1] (the Encyclopaedia Universalis is quite venerated in France :-)) ; I can certainly find several others when I access a library next Monday. Two definitions with two different flavors really do happen in different sources. French Tourist (talk) 10:59, 12 February 2011 (UTC)

By "elementary" I meant middle and high-school text-books. I don't doubt you find many (French, Chinese or English) university-level textbooks (one is cited in the article) which take this path, since it is more smooth and maybe more elegant, too.

More concretely, what precisely do you like to be changed? Mention the integral-based definition in the first section or in the lead? I was reproached (probably rightfully) for having too many internal links in the article, so I'm hesitating about putting a "see below for another definition" in the first section. (NPOV and similar accusations should only be based on assessing the whole article.) I'm slightly more open to putting a brief note in the lead section, but even here it is not clear to me that this would be beneficial. Jakob.scholbach (talk) 14:56, 12 February 2011 (UTC)

My God, I realize only now that while I had of course seen section 4.3 I had read it too quickly and not seen the sentence "The right hand side of this equation can serve as a definition of the natural logarithm". Sorry for the inconvenience I have caused by speaking so lengthily while I had missed something essential. A footnote after the first definition hinting there is another one further might be useful, though, now that this first definition has been reinserted in the article proper (I agree it would be a unproductive in the lead), but feel free to think this is pointless cluttering, and that I was definitely wrong on this one. French Tourist (talk) 17:01, 12 February 2011 (UTC)

No problem. Further comments welcome ;) Jakob.scholbach (talk) 21:33, 12 February 2011 (UTC)

On higher level articles, the reader can be expected to have the experience that a given concept has several equivalent definitions. On such an elementary concept, there obviously exist readers unaware that a mathematical topic can be defined in two seemingly very different ways. So, at the level of this article, my opinion is that I think this is a problem which should be addressed, and which is very difficult to address, especially if we begin to try to imagine the situation of a reader clicking on a chain of three or four successive wikilinks to expand a definition. I have no such constructive solution as for my first remark, but I could suggest that somewhere in the article there should be a subsection titled "Definitions of the logarithm" which will explain to an hypothetical lost reader that some books (with at least an example) do prove from other premises that the logarithm is "the exponent to which the base must be raised". I fully understand this kind of joke can awfully clutter an article, and I would understand quite well if I was rebuked - you might be right indeed. But I think it is necessary to think about it, even if to decide to do nothing. French Tourist (talk) 22:21, 10 February 2011 (UTC)

• Comment I think it would be highly beneficial if this article could explain for the lay reader, up front, the true benefit of logarithms: that, prior to the invention of the computer, logarithms turned a tedious number crunching operation (multiplication and division) into a simple procedure (addition and subtraction). Hence the benefit of logarithm tables and the slide rule. Thanks.—RJH (talk)

I tried to give more emphasize to it. Better/sufficient now? Jakob.scholbach (talk) 19:28, 11 February 2011 (UTC)

Thank you.—RJH (talk) 21:23, 11 February 2011 (UTC)

Brianboulton comments: In an effort to judge the accessibility of this article to a general audience, I read it through - or as much of it as I could understand. It's a while since I studied maths at school, and though I half-remembered some terms I found most of the article beyond me. I enlisted the help of a 17-year-old in his final school year, who is intending to study mathematics at University. His view on the article is that most of his school class would be able to follow the article without too much difficulty, though given the nature of the subject it is unlikely that it can ever be written in terms that the mythical "general reader" would find comprehensible. Between us we picked up a few instances in the lead that we thought could be clarified:-

• Clarify that "e" is a mathematical constant. Don't rely on the link on e
• "It is critical to calculus since it is the inverse function of the exponential function." Rather than relying on links to explain this complex sentence, in the lead I would simplify it to "The natural logarithm is especially critical in calculus", and leave the explanation for later in the article.
• "...the Richter scale is the common logarithm of the amplitude of a seismic event." Unclear as it stands, suggest "...the Richter scale uses the common logarithmn to measure the amplitude of seismic events".

Beyond the lead it will definitely require at least some mathematical knowledege to make sense of most of the content, though often the general gist can be gleaned by non-experts. I hope that this is useful by way of general advice. Please contact me if you need any clarification concerning these points. Brianboulton (talk) 22:46, 16 February 2011 (UTC)

Thanks for your comments, all of which I found useful. They are now integrated in the lead. Jakob.scholbach (talk) 22:00, 17 February 2011 (UTC)

Section on monotonicity properties is needed.

The fact that log_b(x) < log_b(y) ⇒ x < y (with reverse inequality for b < 1) is never spelled out explicitly, although it is quite useful. Similarly, there is log_b(x) < y ⇒ x < b^y (again, for b > 1). In fact, I believe the case b < 1 should be given more attention: although such bases never used in practice, it's a common trap in highschool exams to give such a base and have the direction of monotonicity reversed. Starting from the first picture (with 3 logarithm curves), we should add a 4th curve there, corresponding to, say, b = 1/2. Because right now the article leaves the impression that logarithm is always an increasing function, which is not quite correct.  // stpasha »  02:12, 22 February 2011 (UTC)

Now briefly mentioned (in the inverse function section). As you rightly say, bases < 1 are hardly ever used and highschool exams are not important enough to give a full-fledged discussion of the function, I think. Jakob.scholbach (talk) 21:04, 22 February 2011 (UTC)