Talk:Logarithm

Fractional exponents as discrete logarithms

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

Args in parens

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

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

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

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

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

Footnotes with Google books template

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

unhappy with recent edits

In recent edits, Alsosaid1987 introduced a number of "definitions" of the logarithm. I am seriously unhappy with these edits: these "definitions" are, IMO, (more or less elementary) analytic properties of the log function and should be stated as such (or omitted, depending on the relevance). While it is correct that the notion of exponentiation to real exponents requires elementary calculus, more specifically the concept of a continuous function, these "definitions" are way more complicated (as one sees from the fact that one needs integration, which in particular subsumes continuity in its definitions and proofs of existence etc.)

I suggest to completely remove these "definitions" at this stage. They can be briefly mentioned in the later parts (and indeed the first two already are, the third is not and I don't think it is particularly relevant).

I am also, somewhat less though, unhappy with edits of Purgy Purgatorio: the term b^x does not lack a rigorous definition in whatever generality. It has a completely rigorous definition, which should be the content of our article on exponentiation. (But which should not be the main focus of this introductory section here.) Comments? Jakob.scholbach (talk) 21:41, 30 April 2018 (UTC)

I believe that mathematically/historically it is correct. That doesn't mean that it can't be improved, though. As well as I know it, and maybe not well enough, irrational exponents didn't come until after calculus, and also that ln(x) was first defined through the integral, that is, area under the curve, of 1/x. Much of they way math is taught now is historically wrong, but easier to teach and learn. Among others, integration is more fundamental, as the area under a curve, than the derivative as slope. There are some calculus books that teach integration first, and derivative as the inverse operation later. (Best for students that already know some calculus.) Gah4 (talk) 22:22, 30 April 2018 (UTC)

The focus of this article, as almost any other math article should be on keeping simple things simple, and not on keeping the historical order. (We do have a history section, which has this latter priority.) Mathematical history has nearly always seen a trend to simplifying things, and we don't do the reader any service by presenting the partial / unstructured understandings present in old-day-math. Jakob.scholbach (talk) 21:09, 1 May 2018 (UTC)

In a first step I revised my statement about a lacking definition, to better express my, and hopefully also Alsosaid1987's, original intention. I badly edited his statement, which, I suppose, was intended to motivate the extensions from integer-exponentiation to rational- and finally to real-exponentiation in this ad hoc manner. As I expressed in my edit summary I am not fully convinced about having this section in even this sketchy full g(l)ory, but I certainly do plead for an explicit hint to the difficulties, hidden behind the immediate intuition presented in the beginning of the Definition section.

As regards coining definitions as properties and vice versa, I take a very flexible approach: it depends on which is which, each one taken as definition turns the others to properties (requiring different proofs, of course, and isn't the integral mentioned as the most favored, somewhere?). Considering the average level of knowledge assumed for possible target readers I think it is appropriate to offer this overview of possible definitions exactly in a section Definitions.

In any case, improvements are always possible. Purgy (talk) 06:58, 1 May 2018 (UTC)

Yes, improvements are always possible, but this section and the article as a whole has not improved, in my opinion, by the addition of this material. And I hardly see how to fine-tune these edits so that they do improve the article. Please also keep in mind that this article is a featured article, and has seen a tremendeously detailed FA review (see the logs linked above).

I think the widely agreed structure in math articles is to begin with simple things, and gradually increase depth and width of the article, so to speak. Once again, what is the merit of presenting a "definition" which can not be understood / appreciated without solid foundations in calculus if there is one which avoids this problem nearly completely?

Also, what you are (as far as I understand) attempting to explain here, is not the topic of this article, it is the topic of exponentiation!

I also disagree with the statement that defining logarithms via integrals is a "definition", let alone the most favored one. The fine point that b^x for irrational x needs explanation will not worry 99% of our readers. The remaining 1% should get a short hint "go to exponentiation if you worry / want to know about this". Attempting to create understanding of this 1% by introducing more advanced statements about logs at this point is fruitless, I am firmly convinced. Jakob.scholbach (talk) 21:09, 1 May 2018 (UTC)

my $0.02: I find it to be disingenuous to give a "definition" of the exponential function for integer exponents and "define" the logarithm as the inverse of this function while shoving the problems of defining b^x for real x under the rug. While I agree with Jakob.scholbach's point that most readers won't appreciate the need to have a rigorous definition, Wikipedia needs to give a definition that is correct and accurate.

I advocate the integral definition as the cleanest one. Obviously, it requires the Riemann integral to be defined (i.e., linked), but the point is, there is no clean and simple definition of the logarithm. High school texts routinely "define" log in this essentially circular way. Wikipedia should not repeat the lies (or fairy tales) of schoolbooks. For alternative definitions, I think the prudent thing to do is to move these details later on in the article. While using the inverse definition is okay for motivation, a note should be made that defining the log as the inverse of the exponential function requires a definition of the exponential on the reals and showing that the function is in fact invertible.

I do note that analytical properties of b^x are *not* needed to show existence and uniqueness of the logarithm (see Rudin, p.22, problems 6 and 7).

Alsosaid1987 (talk) 02:35, 2 May 2018 (UTC)