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I have a question concerning this question of mine:

How to prove $f(x)=\ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln(xy)=\ln(x)+\ln(y)$.

The question is simply the following:

  1. why was this question so downvoted?

I feel the actual answer to the problem is far from obvious (at least for me, and it took me a bit to see all the steps behind the accepted answer – as can be noticed by the comments below it). Moreover, I do not see what I should have added to the text to make it more clear.

True (as can be seen by the recent edit), there where two problems in my original question:

  1. I did not mention what had to be proven (i.e. that $f(x) = \ln (x)$ is continuous);

  2. I used $\log$ instead of $\ln$ (which actually comes from my problems with logarithms...).

[Mind that, from the comments of two users below the question, I edited the original question few minutes ago accordingly, after having realized the problems]

Thus, to expand the original question a bit:

  1. is the case that the question was downvoted due to these two points?

  2. If that's the case, wasn't it better to comment (as the users did) before downvoting?

Just to explain one point, this metaquestion does not come from the fact that – so to speak – downvotes imply losing points. It's just that this site is extremely helpful for people like me, without any possibility to formally learn math.

As a matter of fact, I would like to understand what happened, because I would like my questions to be as efficient as possible for all those people that are kind enough to spend their time helping me understand.

As always, thank you for your time.
Any feedback will be greatly appreciated.

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    $\begingroup$ Only the downvoters could tell. Regarding your listed problems: Ad 1., that's not quite true. You forgot to state it in the title at first, but it was clearly stated in the body from the beginning. Ad 2., that's not a problem at all. For most mathematicians, $\ln$ is just $\log$ spoken with an engineer's accent. The comment stating the importance of the definition of $\log$ supposedly means it is important for the way how to prove continuity at $1$, and refers to whether $\log$ is defined as the inverse of $\exp$, or as $\log x := \int_1^x \frac{dt}{t}$ or so. $\endgroup$ – Daniel Fischer Jun 7 '16 at 17:47
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    $\begingroup$ For how $\log$ is defined is not relevant for how to deduce the global continuity from the continuity at $1$ together with the functional equation $\log (xy) = \log x + \log y$. $\endgroup$ – Daniel Fischer Jun 7 '16 at 17:49
  • $\begingroup$ @DanielFischer thanks a lot for the technical feedback. Just to be sure I got the point, if $\ln$ things work smoothly in the derivation of the continuity at $1$, because it is the inverse of $\exp$, while in the other case it is just more cumbersome. Still, at the end of the day, it does not really matter because it works for whatever base we choose for the $\log$. $\endgroup$ – Kolmin Jun 7 '16 at 18:24
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    $\begingroup$ It's not that proving continuity at $1$ would be easy with one definition and hard with others. You need totally different arguments for the different definitions. Since logarithms to different bases just differ by a constant factor from each other, the base of the logarithm under consideration doesn't influence the structure of the proofs, just minor details (a constant factor on some terms in the inequalities). $\endgroup$ – Daniel Fischer Jun 7 '16 at 18:39
  • $\begingroup$ Thanks a lot for the clarification. $\endgroup$ – Kolmin Jun 7 '16 at 19:04
  • $\begingroup$ Just to clarify I had asked to provide a definition of $\log x$ but I did not downvote. As a rule I downvote not for lack for information, but for presenting incorrect/false information. And like many users of MSE I am also frustrated when downvotes are not accompanied by some comment from downvoters. $\endgroup$ – Paramanand Singh Jun 11 '16 at 21:14
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It is difficult to guess why other users voted the way they did. But let me try anyway.

The question has, at the moment, one close vote and the chosen reason is: "This question is missing context or other details: Please improve the question by providing additional context, ..." If you follow the link, you can find more about what is meant by providing context.

One way of doing so could be showing what you have tried. Or you can add the origin of the problem. For example, you write in your post: "I read that this can be proved in two steps." Maybe you could include information where you read this.

Additionally, I can see two ways of interpreting the question (even in its present form).

  • Use the fact that $\ln x$ is continuous at $x=1$ together with the identity $\ln(xy)=\ln x+\ln y$ to show that logarithm is continuous everywhere. (I.e., the continuity at the point $1$ is given.)
  • Prove first that $\ln x$ is continuous at $x=1$ and then use this fact to prove that it is continuous everywhere. (I.e., the continuity at the point $1$ is supposed to proved.)

If it is the latter, then the question is difficult to answer if you include what your definition of $\ln x$ is.

I should also say that I consider using $\log x$ instead of $\ln x$ as an unlikely reason for the downvotes, since $\log x$ is commonly used to denoted the natural logarithm.


Maybe some people would consider these points too minor to be a reason for a downvote. But different users of this site have different standards for upvoting and downvoting, closing and reopening.

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  • $\begingroup$ I do not think it is very relevant, but I will mention that I did not vote either way on the question. Also, at the moment you have 4 upvotes and 3 downvotes on the question, which means the net profit of $20-6=14$ reputation points. $\endgroup$ – Martin Sleziak Jun 7 '16 at 17:52
  • $\begingroup$ Thanks a lot for the answer! In particular, it is quite useful what you point out concerning the ambiguity of the question itself. I would say this ambiguity is built-in in the question, in the sense that sometime a question is ambiguous because it is ambiguous for the OP, and it signals it. Related to this, I will add where I read the statement (it is in a previous comment on the site). $\endgroup$ – Kolmin Jun 7 '16 at 18:13
  • $\begingroup$ Concerning the net profit, as I wanted to point out, this is not really the issue. Here there are a lot of (supernice) people that are willing to help users like me with their problems: I simply want to ease their task in case they actually decide to put an effort to give me an answer or a comment (of course in doing so, I also want to maximize the possibility of getting an answer). That's what this question is really all about: to know how to produce clear questions. $\endgroup$ – Kolmin Jun 7 '16 at 18:16

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