Talk:Logarithm

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[edit] Merging of Natural Logariths with logarithm#unspecified bases

In the natural logarithms article Natural_logarithm it is suggested that this article be combined with the logarithm#unspecified bases article. I am totally against this idea, and would like to know why somebody suggested it. Unless there is a strongly compelling reason to merge, then I suggest that this notice, which links to this page, should be removed. --New Thought 20:48, 25 August 2005 (UTC)

Obviously I referred to the section "Notational conventions", not the whole article, of course. --Army1987 21:21, 25 August 2005 (UTC)

[edit] Notation

  • ln = logarithm to the base e
  • lg = logarithm to the base 2
  • log = ambiguous (engineers: base 10; mathematicians & scientists: base e; computer scientists: base 2)

ln is universally recognised, so it should stay. Whenever log is used, the base should be specified.

The above seems to be a good summary of what was here before.

And I agree with the above, but I think that lg should not be used (even if it is the correct notation) because it is too rare. I myself have read some calculus books that say 10 should be the "default" base, and some that say e should be the "default" base.

Brianjd 06:11, 2004 Jun 19 (UTC)

For the old discussion of notation, see Talk:Logarithm/Notation. - dcljr 23:32, 8 Nov 2004 (UTC)

Two things: (1) For me, `lg' is just as ambiguous as `log'. Log base 2 can be referred to unambiguously (I hope) as `ld', but even that isn't very common in areas where base 2 is dominant (computer science, information theory). (2) Specifying an explicit base for `log' is a good idea in many cases, but there are situation when this is not necessary. For example, in computer science O(log n) is unambiguous, since the choice of base does not matter. MarkSweep 08:38, 2004 Jun 20 (UTC)

Most UK A-level textbooks I've seen like to use lg for log10 and ln for loge (the rest use log for log10, and I've never seen lg used for log2). Style is up to the article writer, but I propose that, if only one base is used in the article, the first occurrence of log includes the base. Any further occurrences don't need to include the base, since the default base is implied. But, really, we don't mean sin(x°) when we say sin(x), because measuring angles in radians is much more useful (namely, you can use the Taylor series, and sin'(x) = cos(x)). Similarly, log'(x) = 1/x. If we define log = log10, then we're stuck with log'(x) = 1/(x ln 10), which is ugly at best. Of course, log(2) is another special number (namely, it's important when calculating logs of big numbers). --Elektron 15:30, 2004 Jun 30 (UTC)

Forgive my impoliteness, but let's stop wasting time here. A foolish consistency is the hobgoblin of little minds. Here, people seem to be under the impression that a new symbol needs to be created for each base in common use. Otherwise, how else will the moronic readers tell the bases apart ?

That strikes me as strange. Shannon didn't seem to need "lb" or "lg" or anything like that as a crutch for his readers. His readers just assumed from the context (information theory) that, by "log", he meant the binary logarithm. Somehow, they were able to use context clues to figure out which meaning of the word was in use, just like every other word with multiple definitions.

It also strikes me as strange that, in the rare situation we'd need to be clear about a base, we wouldn't use the clear logbase. Instead, we'd irrationally use abbreviations, which are unclear and used for common, well-repeated situations.

Let's treat log, its abbreviation lg, and its capitalization Log like any other word with multiple meanings and stop arguing over definitions. Thank you. HAND. — 131.230.133.185 08:09, 14 August 2005 (UTC)

But we should say what notations are, not what they ought to be. As for me, I use ln for natural logarithms, log for indeterminate logarithms, and always specify the base otherwise. (If it is obvious from context, e.g. 2 for Information Theory or musical intervals, I could use simply log, but I'd however specify what the base is once and for all at the beginning of the article.) But the reader should be informed that other people use lg to mean log2 (that's written even in natural logarithm article, and no doubt it belongs here more than there) and Log for log10.--Army1987 16:10, 14 August 2005 (UTC)

I agree with mentioning in the article that lg is sometimes used for log2. However, my problem is that after your edit, the article claimed that the notations ln, lg and Log are unambiguous. Well, ln is indeed unambiguous, but lg is ambiguous (can also mean natural or indefinite logarithm) and Log is also ambiguous (can also denote the logarithm as a multi-valued complex function). -- Jitse Niesen (talk) 17:05, 14 August 2005 (UTC)
Yeah. You're right. As for lg I messed up a little, and after my edit the article wasn't even self-consistent. As for Log i wasn't even aware that it had any other use than log10...--Army1987 21:16, 14 August 2005 (UTC)
The article already mentions that "lg" (as well as "log") is used frequently in certain subjects to refer to base two. The simple and concise guiding principle of looking at the context will serve readers new to logarithms at least as well and probably far better than a long listing of hemming and hawing and apologizing over badly applicable tips and tricks. Edits to repeat what was already in the article and other edits ("Log" = log10, etc.) simply serve to crowd the article with the "it's used as ... but not really, because sometimes its used as ...". Concise advice that nearly always works is always better than long-winded, detailed advice that works about half the time.
As you noted, you know exactly what you mean by certain notations ("As for me, I use...") when you write them. You also know exactly how you've experienced them in other works. That is the problem: you (along with me and the rest of us) are nowhere near omniscient. The longwinded version invites people to add their own hasty generalizations to the article, leading to edit wars, and, finally, to the compromise monstrosities that result when trying to include ten people's viewpoints (as already seen in the edit history).
For those that must know how binary logarithms are commonly written, the common shorthand "lg" is already mentioned in the binary logarithm article. In fact, with the links to the detailed articles for each frequently-used logarithm, it should be easy for those who want more information (or want to add detailed information) to go to those articles.
The people doing detailed research on usage will go to the four in-depth articles, anyway. The people new to logarithms, looking to get started using and reading them, don't need to be awash in details from everyone's attempt to show off their knowledge in a prominent place. This is an overview article, after all. — 131.230.133.186 00:24, 16 August 2005 (UTC)
Yes. But you should've fixed the common logarithm page to reflect the change. (Now I did that.) And notice that the note about ln is still there (though I admit ln for loge is way more common than lg for log2 or Log for log10. However, at least one sentence for them could be retained...--Army1987 09:47, 16 August 2005 (UTC)

[edit] Definition

I thought:

  • x^(r/s) = (the rth root of x)^s = the rth of (x^s).
  • The logarithmic function is any function which satisfies four conditions:
    1. Domain is all of R.
    2. It is nonconstant.
    3. It is differentiable.
    4. For any real numbers x, y, L(xy) = L(x) + L(y).
  • The natural logarithm of x, denoted ln x, is the definite integral of 1/x with the lower limit 1 and the upper limit x.
  • The exponential functions are then inverses of the logarithmic functions. This is the only way to define exponentials for irrational exponents.

I have not seen anything like this on Wikipedia. Only that e = lim (n->infinity) (1+1/n)^n and logarithms are inverses of exponents, which is absolute rubbish according to what I have read elsewhere.

Brianjd 08:25, 2004 Jun 18 (UTC)

That anything is the only way to define exponentials of irrationals is surprising, to say the least. Seldom, if ever, is there only one way to do something in mathematics. The power-series
\exp(z)=\sum_{n=0}^\infty{z^n \over n!}
defines exponentials, at least to base-e, for all complex numbers, including irrationals. If you want other bases, you may want to bring in already-defined natural logarithms, and then perhaps you want to proceed as you indicate above, but any assertion that that is the only way is dubious.
Contrary to your assertion, the domain of the natural logarithm function is not all reals, but only positive reals. There is a "multiple-valued" (if that expression can be forgiven) logarithm function defined for all complex numbers except 0. But how did you propose to define the logarithm of 0, if the domain is all reals? Michael Hardy 21:39, 18 Jun 2004 (UTC)

What I have read seems to imply that defining exponentials that way would be seen to be, by most people, as absurd as defining pi by a power series.

Yes, the domain of the natural logarithm is all positive reals. My sources were correct - I posted incorrectly.

Anyway I have not read Wikipedia pages very thoroughly but I remember seeing basically "logarithms are inverses of exponentials; exponentials are inverses of logarithms". Is there any solid definition?

Michael Hardy posted the same response to my talk page. Can we keep the discussion where it belongs?

Also, don't post to someones talk page with the section heading "Definition". When things are taken out of context like that they become gibberish.

Brianjd 05:52, 2004 Jun 19 (UTC)

Exponentials at least to the base e seem to be defined properly...but the proofs seem very complicated.

I have seen a calculus book with much easier proofs of the product rule, quotient rule, and the various proofs related to logarithms/exponentials (logarithms require some knowledge of integration though; as the book used my definition above).

Brianjd 05:55, 2004 Jun 19 (UTC)

I see that my definition is indeed buried in the middle of the natural logarithm page. I think that all common definitions (and mine qualifies as a pretty common definition) should be listed in the intro, or in the first section, with their equivalence proved. Brianjd 06:01, 2004 Jun 19 (UTC)

Don't define the logarithm. As with anything in math, there are too many equivalent definitions to call any of them the definition. Rather, list the properties of the log function, with a side note that they are equivalent. Also, no need to prove this in an article.--Sean Kelly 20:05, 24 Dec 2004 (UTC)

e is just a number that satisfies a certain property. From first principles (with shamelessly incorrect use of dx),

\frac {d}{dx} a^{x} = \frac {a^{x+dx} - a^{x}}{dx} = \frac {a^{x} a^{dx} - a^{x}}{dx} = \frac {a^{dx} - 1}{dx} a^{x}

So we define e as the number such that the first derivative of ex is ex. That is,

\frac {e^{dx} - 1}{dx} e^{x} = e^{x}
e^{dx} - 1 = dx\,\!
e^{dx} = 1 + dx\,\!
e = \left(1 + dx\right)^\frac{1}{dx}

If we define dx = 1/n,

e = \left(1 + \frac {1}{n}\right)^{n}

Alternatively, if we let f(x) = ex, it's trivial to prove by induction that

f^{(n)}\left(x\right) = e^{x}

Which is where you get the taylor expansion for ex

The basic definition is, really, just d/dx ex = ex. I believe I've just proved that the standard definitions of e are equivalent. --Elektron 16:17, 2004 Jun 30 (UTC)

Um, could someone put the basic formula of a logarithm in big letters somewhere? Like (but using variables) log(base 2)8 = 3 and 2^3=8 are equal. I know that example is used in the article, but it would be great if a more general one was included that used the big math font. --pie4all88

You mean,
\log_b x = n \qquad \mbox{means} \qquad b^n = x
? Also, it shouldn't be called the "basic formula" but rather one way of algebraically defining it.--Sean Kelly 20:05, 24 Dec 2004 (UTC)

[edit] Formula moved from article

I moved the formula

\sum_{j=1}^{k-1}\left( \frac{-2^{-1 + j}}{\left( -j + k \right) \,\left( -1 + k \right) !} \right) + \frac{2^{-1 + k}\,\ln (2)}{\left( -1 + k \right) !}=\sum_{n = 0}^{\infty }\frac{{\left( -1 \right) }^n\,n!}{\left( n + k \right) !}

here, because it is not clear to me what it is doing at that place in the article. What connection does it have to the algorithm? Furthermore, I'd like to see a reference (I'm too lazy to check it myself). -- Jitse Niesen 14:55, 17 May 2005 (UTC)

[edit] vagueness in need of fixing

This article says:

Some people use the notation: blog(x) instead of logb(x).

Since I've never seen this and I've been around, I'm guessing that the phrase "some people" could be made more precise, e.g., "Researchers in mathematical omphalology use the notation blog(x) instead of logb(x)." Could anyone who knows who uses this notation add that information? Michael Hardy 21:58, 22 July 2005 (UTC)

I have seen this notation in Dutch maths text books for secondary schools, see for instance here (Dutch PDF file, look at pages 6 and 7). However, I'd rather not put that in the article myself since I have really no idea how wide spread it is. -- Jitse Niesen (talk) 23:16, 22 July 2005 (UTC)

[edit] Infinity

I vaguely recall from grade school that the log of infinity is infinity -- i.e., it never approaches a specific number. Maybe adding this little tidbit could be helpful, I don't know.

I've added a link to table of limits. This could be added there. --MarkSweep 21:24, 2 August 2005 (UTC)


[edit] Napier's bones

Napier's bones are not related to logarithms, whereas slide rules clearly are. I interpreted the relevant sentence to refer to multiplication using logarithms, rather than methods for multiplication in general. Perhaps this should be made clearer. --JahJah 18:58, 19 August 2005 (UTC)

Jitse Niesen, I appreciate the rewrite. However, slide rules use logarithms for multiplication, eliminating the use of tables, while Napier's bones are unrelated, except throught the name. One could equally well mention using the abacus here, as well as many other paper and pencil techniques. --JahJah 19:41, 19 August 2005 (UTC)

True; feel free to improve (I misinterpreted your first comment). What I did not like about the previous formulation ("Napier's bones and slide rules made work even easier: tables weren't necessary.") is that slide rules cannot fully replace tables of logarithms since they are less precise. -- Jitse Niesen (talk) 20:16, 19 August 2005 (UTC)
I have rewritten the passage taking your comments above into account. By the way, I am old enough to have used slide rules, log tables and mechanical calculators when I was in high school. --JahJah 20:43, 19 August 2005 (UTC)

[edit] Antilogarithm

Anyone have a reference for the comment that the use of antilog was never widespread? It certainly was common in books of tables when I was in high school in the 1960s.--JahJah 10:37, 20 August 2005 (UTC)

The OED has a citation for antilogarithm (in the sense of this article: it can also mean other things) from 1675. What's the source for the 1800s in the article. --JahJah 10:56, 20 August 2005 (UTC)

[edit] recursion algorithm for logarithm

I don't understand how to get the higher n terms in the expression log(x)=n0 +1(n1+1/(n2+1/(n3+... if I use the same method as I did to get n0, all the higher n terms will be equal to 0. (Forgive my ignorance, but I asked people with degrees in mathematics and it's not so transparent to them how to get the higher terms).

[edit] programming languages

I've added a bullet on programming languages to the Logarithm#Unspecified_bases section. It's from memory; someone please check it for accuracy (and whether Pascal should be included). Any corrections should also be applied to the corresponding bullet at Natural_logarithm#Notational_conventions. Thanks, Trovatore 05:13, 9 September 2005 (UTC)

Confirmed for Java [1] Lee S. Svoboda tɑk 18:45, 18 May 2006 (UTC)

[edit] rm "clear notation" section

I've removed the following claim:

In order to avoid confusion, a logarithm should sometimes be written to make its argument explicit:
\!\, \log_b(x) + z \mbox{, rather than } \log_b x + z
This avoids confusion with:
\!\, \log_b(x + z)

In fact "logbx+z" is perfectly clear and unambiguous, and means the same as "logb(x)+z --Trovatore 16:30, 10 September 2005 (UTC)

In fact I once changed it to:

In order to avoid confusion, a logarithm should sometimes be written to make its argument explicit :

\!\, \log_b(x + z) \mbox{, rather than } \log_b x + z
This avoids confusion with :
\!\, \log_b(x) + z
Which is what IMO that was most likely to mean. But MarkSweep reverted my edit.--Army1987 20:50, 11 September 2005 (UTC)

[edit] Complex Logarithm

I'm missing remarks on the complex logarithms (different branches etc.). However, I dont feel mature enough in that subject (yet), so I decided not to insert anything regarding this issue. Seraph85 14:26, 20 December 2005 (UTC)

[edit] Useless?

Amazing how one can write this much info in an online, public and hopefully educational to the masses - in such a fashion that only those who understand the article are those who could have written it to begin with.

I've no doubt this info is terribly mathematically accurate.

It's also terribly useless - I still have no idea why log is important or a reasonable example of it's use that a non-mathemetician could use.

What a waste of WIKI space.

I can buy math books. I don't come here for such info. —The preceding unsigned comment was added by 24.43.125.165 (talk • contribs) .

It depends what your expectations are. Logs are not as important for everyday use as say addition and multiplication. One runs into them if one does numbers a lot, like an engineer, statistician, or somebody working in programming or in financial markets. If you paint landscapes for living, this page will be useless to you no matter what. Otherwise, you are welcome to give constuctive suggestions. Oleg Alexandrov (talk) 17:57, 10 December 2005 (UTC)
I am a student of Electrical Engineering and Computer Science and found this article to be useful. Robin.Anderson 23:30, 24 March 2006

Almost all the wiki articles i've seen on mathematical subejcts are like this - they are fine if you already know what the subject in question is about. Reading this, I haven't any idea what a logarithm is, what it might be used for, or how on earth to use one myself. --anon

You could start by making an account (each time you visit you have a different IP address). Then you could learn how to sign your posts (like this: ~~~~) I would be interested in talking to you, but you should do the two above and also visit this page more often. So far your attitude is unhelpful anonymous infrequent complaining, and I will not spend time on that. Oleg Alexandrov (talk) 01:44, 14 December 2005 (UTC)

yeah i think this needs to be simplified so people like me can understand it. i came to this page trying to figure out how to solve a problem with moore's law, and this article was no help. all i wanted to know was 1*10^9=2^[(y-2003)/2]*8*10^7 ; solve for Y. I still cant figure out how to write that with Log on my calculator. J C 03:59, 30 January 2006 (UTC)

Your problem is about solving exponential equations. Take the log on both sides, and then this article will become useful, as you will need to use the log properties, like the log of product, the log of power, etc.
I don't think it is fair to blame the article for you not being able to solve the problem. This article is (and should) describe the properties of logs, and is not a textbook chapter on solving exponential equations. Oleg Alexandrov (talk) 21:11, 30 January 2006 (UTC)
I think it might be worthwhile to take a stab at http://simple.wikipedia.org/wiki/Logarithm. Well, I'm not going to, but that would seem to fill the void that some people see. -Rjyanco 00:38, 31 January 2006 (UTC)

[edit] Work needed?

(continuation of prev discussion; this comment is by Oleg Alexandrov (talk) 04:45, 1 February 2006 (UTC))

That article is yet to be written. I think the article here does need work.--agr 11:59, 31 January 2006 (UTC)

I would like to ask for specific suggestions of what needs work, before work is attempted. Just to make sure there is understanding on what is going on before committing a lot of time and effort. Oleg Alexandrov (talk) 01:38, 1 February 2006 (UTC)
Sorry, I went ahead and did some serious editing. Mostly I reorganized things, with the intro focusing on the basics. I moved the continued fraction algorithm and the proof of the basis change formula to additional topics in logarithms, which I renamed from "using logarithms", a somewhat misleading title. I also tried to make explanations clearer in some places. I think the section on tables needs more work and maybe should be in its own article. --agr 04:06, 1 February 2006 (UTC)

The table reflects the most important properites of the log. While a separate article on that with more details may be needed, I would be against removing the table from the article. Oleg Alexandrov (talk) 04:45, 1 February 2006 (UTC)

I was referring to the subsection "Tables of logarithms", not the table of formulas, which I agree should stay here. I think there is a lot more that should be said about tables of logarithms and that might justify a separate article. Sorry I did not make myself clear. --agr 05:16, 1 February 2006 (UTC)
I see. A common practice on Wikipedia is to create a main article, say "Tables of logarithms", put there a lot of stuff, and then have a shorter version of that at logarithm, while referring to "table of logarithms" for more details. How would that sound? Oleg Alexandrov (talk) 23:52, 1 February 2006 (UTC)

[edit] changes needed

The article needs to be divided into two parts - the first part must give an explanation FOR NON MATHEMATICIANS. Mathematicians will not be looking in an encyclopedia to find out about these subjects.Johncmullen1960 05:46, 25 October 2006 (UTC)

I second this wholeheartedly. The concept of logarithms is not so esoteric that the introduction can't be expressed in plain English for the layman. Later sections can dive into as much detail or complexity as desired. --Ahhwhereami 07:37, 28 February 2007 (UTC)

[edit] Jainas?

I'm suspicious of the comment about the Jaina mathematicians at the beginning of the history section, since it's uncited anywhere in the article, and since there has been some controversy recently over overzealous crediting of this group. Is there someone who could clear this up? Ryan Reich 15:02, 22 April 2006 (UTC)

[edit] New casio

A new casio calculator( The Casio FX83ES - Scientific Calculator) can calculate log's of any base. so there is no need for conversions. page on the webiste.-wolfmankurd

[edit] Changed the program

This program is easier to understand and faster than the previous one.Wolfmankurd 20:51, 27 April 2006 (UTC)

Sure your program is easier to understand and faster BUT

* How do you calculate 3.5**4.5  without using logarithm?

Also your program does not work! Have you tested it yourself?

$ python prog.pl
Log Base: 2

ERROR!

Ohanian 01:03, 8 May 2006 (UTC)

[edit] Definition

The article said that the exponent, or logarithm, indicated how many times to multiply a number by itself. However, an exponent of one does not mean to multiply it by itself one time. That would be an exponent of two. So, I have changed the article to indicate that a positive exponent indicates that a number is multiplied by itself a number of times, with the exponent indicating how many times it's used as a factor. This is more accurate. It leaves the problem of a positive exponent of 1 still being characterized as meaning a multiplication using the number in a multiplication a number of times, in which the number is used only once in the multiplication - an awkward element of this definition, since there is no multiplication at all if the exponent is one. However, it could still be interpreted as meaning it is used as a factor only once (i.e. no multiplication), and I did not judge that complicating the definition with an extensive discussion of this exception was worthwhile - but rather left it to this note to point this out - what is still missing in this improvement in the definition. If someone sees a simple way of describing the exception that fits well in the context, I would welcome its being added. One possibility would be to indicate that the definition is for positive integers greater than one. This might be a good solution. -- Ken Cliffer

I tried to write it as "product of n factors with each equal to b". How's that? :) Oleg Alexandrov (talk) 02:59, 14 May 2006 (UTC)

Oleg, thanks for your input. Please excuse that I'm leaving it to you to make the change, since I think it's better that you agree before I keep changing it (I'm new to this, so please let me know if I should just make the change and comment on it). My comment and suggestion: I like your previous version better than the current one. However, even better, my suggestion would be to change it to this (a little shorter than the previous one, but better still than the current one): "... the product of n factors equal to b ...". I think this is better than "... the product of n copies of b ...," since "copies" seems to imply something more than that it is simply numbers - there's nothing "copied," only factors equal to b. [I had another suggestion here, but on further consideration, I realize I was missing something; therefore I have withdrawn it.]

Okay, I have now read guidelines to be bold (after your invitation to join), so I'll go ahead and make the suggested change.KCliffer 20:59, 14 May 2006 (UTC)

Cool, thanks! Oleg Alexandrov (talk) 00:10, 15 May 2006 (UTC)
I think the current thing about factors is slightly confusing. If you need a definition of b^n, that definitions going to be confusing as all hell. I propose:
"If n is a positive integer, bn means b multiplied by b, n number of times."
I think even this definition doesn't suffice because I don't think a concept of "multiplying sumthing a negative amount of times" really exits - not to metion those needing that definition won't have that concept. Fresheneesz 23:08, 29 May 2006 (UTC)
Your definition suffers from the problem mentioned above. What does it mean to multiply n times? b^2 is b multiplied twice? Does that mean two b's or two multiplications? Oleg Alexandrov (talk) 03:22, 30 May 2006 (UTC)
Hmmm, I guess this is why english isn't used to describe mathematics. Maybe we should forgo the english alltogether, as it seems sufficient explanation has been taken from the article exponentiation. Fresheneesz 17:20, 30 May 2006 (UTC)

[edit] Numerical value program

Could somebody write why and how this program works? Not all of us are that familiar with Python. I'd do it myself if I knew Python or was willing to learn it. Lee S. Svoboda tɑk 18:53, 18 May 2006 (UTC)


Sure which do you prefer? Pseudocode or C? Here it is in pseudocode

#
# Function in pseudocode
#
function log(float N,float X)
{  epsilon = 0.000000000001
   integer_value=0
   while (X < 1)
   {
      integer_value = integer_value - 1
      X = X * N
   }
   while (X >= N)
   {
      integer_value = integer_value + 1
      X = X / N
   }
   decfrac = 0.0
   partial = 0.5
   X=X*X
   while (partial > epsilon)
   {
      while (X >= N)
      {
         decfrac = decfrac + partial
         X = X / N
      }
      partial = partial / 2
      X=X*X
   }
   return (integer_value + decfrac)
}
I can take both (Java, my programming language is a C-like languuage). However, I didn't state myself well. Do you know the mathematical formula that makes this program work? Lee S. Svoboda tɑk 16:23, 23 May 2006 (UTC)

[edit] Formal definition?

Since it isn't included, I'm going to guess that this is the formal definition:

\log_a x = y \iff a^y = x

Or, verbosely: log base a of x equals y if and only if a to the yth power equals x.

Can anyone verify this? If so, anyone can go ahead and add it to the article upon verification. -Matt 05:44, 20 May 2006 (UTC)

This is a good way to understand the logarithm, but it's not usually used as the formal definition from first principles. The standard excuse for why it's not used is that defining exponentiation is slightly tricky for irrational exponents, though it's my opinion that this excuse has no teeth. The usual formal definition is as an integral. -lethe talk + 23:14, 29 May 2006 (UTC)

[edit] Recent trimming

I removed some stuff from this article, because I think the Python code is unhelpful to anybody not knowing python (a simple explanation on how the algorithm works and when it works would do a much better job). I also removed the recently inserted series expressions as I feel they were not so relevant, and the article was getting too big anyway. Wonder if there are nay comments on that. Oleg Alexandrov (talk) 17:42, 23 May 2006 (UTC)

I would have preferred the python program replaced by a mathematical formula, but do not disagree.
Lee S. Svoboda tɑk 20:25, 25 May 2006 (UTC)

[edit] Manual Calculation

Does any one have any information on how the vast tables of logarithms were calculated. Clearly, it is not a simple task or tables would not have been necessary. 68.6.85.167 21:52, 29 May 2006 (UTC)

It's nice to see someone has the same doubt. I'd suggest to add at least a line on how those logarithm tables were calculated, they certainly weren't made using a calculator, and else than using one of those, I have no idea on how to manually calculate a logarithm, neither I have heard an explanation of it, ever. Pentalis 00:24, 14 July 2006 (UTC)

[edit] Base of log, ld, lg, ln

The anon editor 87.193.43.142 indicated that ld should be base 2 and lg base 10. He may have a point. The d then stands for dual. I have never seen lg used for base 2.

The section about the various conventions is quite confusing. It might be condensed along the following lines. The abbreviation ln is universally used to stand only for base e and log10 for base 10. The name log can be base e (maths) or base 10 (engineering, including calculators and computer languages). The name ld is sometimes used in computer science for base 2. The name lg is sometimes used for base 10. −Woodstone 19:39, 11 August 2006 (UTC)

The name log can be base e (maths) or base 10 (engineering, including calculators and computer languages). This is wrong. I've never used a programming language where log means anything other than the natural logarithm. Fredrik Johansson 20:22, 11 August 2006 (UTC)
Where do you find that in the article? All I see is the part that says that in most commonly used programming languages, "LOG" means natural logarithm, and the part that says that on calculators "LOG" means base-10 logarithm. Michael Hardy 23:16, 11 August 2006 (UTC)
I was quoting Woodstone. Fredrik Johansson 23:54, 11 August 2006 (UTC)

[edit] Dubious notation

The article has this:

  • Also frequently used is the notation blog(x) instead of logb(x).

I have never seen that notation, and I've been around. Who uses it? If it prevails in some particular field, the article should say which field. Michael Hardy 23:17, 11 August 2006 (UTC)

I haven't seen it either (though admittedly I haven't been around :-). By the way, the symbol logb x can also mean a b-foldly nested logarithm (MathWorld). This should perhaps be covered in the article. Fredrik Johansson 02:13, 12 August 2006 (UTC)
Whoops! I just happened to look in one of my old calculus textbooks and it does use the notation blog(x). How strange that I didn't remember that. This presumably means that the notation is common in Europe. Fredrik Johansson 08:54, 20 August 2006 (UTC)
I had a similar but opposite experience. Before looking at this wikipedia article, I could not remember ever having seen the logb notation. But a quick look in Google confirms it's the dominant notation. In my high school and university training we always used the blog notation. One slight advantage of that notation is the elegance of the formula for change of base: {}^a \log b = {}^a \log c \cdot {}^c \log b. −Woodstone 09:19, 20 August 2006 (UTC)

Frederik, which calculus book are you referring to? Woodstone, where the high school and the university you refer to? Michael Hardy 18:30, 22 August 2006 (UTC)

If it helps, Persson & Böiers, Analys i en variabel. (In Swedish.) Fredrik Johansson 20:27, 22 August 2006 (UTC)

can someone explain where this equation is from....Log a^b in base a^c = b/c...in example log 6^5 to base 36....would mean log 6^5 to base 6^2 which therefore with the equation would result in the solution being 5/2...which is 2.5. -By Babak S. yaaa diiiig

(a^c)^(b/c) = a^(c*(b/c)) = a^b QED. --agr 04:28, 22 August 2006 (UTC)

[edit] History is unreferenced

Th whole of the history section is unreferenced. It's like 'take-my-word-for-it'. It's not encyclopaedic, please reference.

[edit] History section

I've removed an entire paragraph in the history section regarding the contributions of Jaina and Muslim mathematicians. There does not seem to be a source for this at the moment. In some sense, I think what was written falls in the category where we should have zero information for the moment. See ([2][3][4]) for some background. In the meantime, I'm going to see if I can get in touch with a historian and perhaps find some resources to support the claim - I think trigonometric tables were tabulated quite early in the history of mathematics, but I am not so sure about logarithms. --HappyCamper 01:01, 7 September 2006 (UTC)

[edit] .

A great way to explain something so that most people will not understand it. —The preceding unsigned comment was added by Wndwshppr (talk • contribs).

Could you be specific? Michael Hardy 20:14, 31 October 2006 (UTC)


[edit] Johannes Kepler

When talking about the history of logarithms, you cannot exclude Johannes Kepler, Because he was the first to derive logarithms purely mathematically. This unsigned comment was added by 71.195.204.65 23:51, 14 November 2006 and later modified by somebody else.

[edit] Computing section

When discussing how to compute logarithms it might be a good idea to explain how ocmputers compute log(x) - logarithm with base e.

The basic outline can be as follows:

First, the number x is typically a floating point number of the form m * 2^n where m is mantissa and n is an integer exponent.

log(x) = log(m) + n * log(2)

log(2) is a constant - can be computed once and for all.

What is left is to compute log(m) of a relateively small number m.


Should probably also here add in how the series for logarithm is developed.

log(1 + z) can be written as the integral of 1/(1 + z) and is the integral of the series 1 - z + z^2 ... (include a link to power series here) and so log(1 + z) = z - 1/2 z^2 + 1/3 z^3 etc...

Similarly, log(1 - z) = -z - 1/2 z^2 - 1/3 z^3 etc...

If you were to add those two you would simply get log(1+z)(1-z) = log(1-z^2) = -z^2 - 1/2 z^4 etc which isn't very interesting since you get the same if you replace z with z^2 in log(1-z). However, if you instead subtract them you get:

log(1-z)/(1+z) = 2( z + 1/3 z^3 + 1/5 z^5 ...)

Thus, if you can find an x such that (1-z)/(1+z) = m you will be able to compute log(m) by that series.

I.e. 1 - z = m + mz or z = (1-m)/(1+m). If m is very small as for denormal numbers the divisor here is close to 1. For normal floating point numbers you can adjust m to be in the range 1 <= m < 2 or 1/2 <= m < 1 and in that case you will get a fairly low value for z which allows you to compute log(m) fairly rapidly using only a few terms in the computation unless you want very high precision.

Once log(m) is computed - and log(2) can also be computed in this manner - you can then compute log(x) = log(m) + n * log(2) by a simple multiply and add.

log10(x) can then be computed using the formula, log10(x) = y means that 10^y = x and taking log() on both sides gives: log(x) = y * log(10) so log10(x) = y = log(x) / log(10) or in general log(x,b) = log(x) / log(b). where log10(x) = log(x,10) and log(x) = log(x,e). log(b,b) = 1 for any valid base b.

salte 12:53, 1 December 2006 (UTC)

Building on what I wrote on Friday I suggest we include something like this in the main page.

[edit] Computation of logarithm in computers

As all other logarithms can be computed based on any other, it is natural to first define the logarithm for a specific base. Using e as this base one can then easily compute other logarithms based on that function.

log(x) can be computed using a series. This series is found by first looking at

\frac1{1 - z} = 1 + z + z^2 + z^3 + z^4...

Integrating on both sides gives:

-\log(1 - z) = z + \frac12z^2 + \frac13z^3 + \frac14z^4 + \frac15z^5...

or

\log(1-z) = -z -\frac12z^2 - \frac13z^3 - \frac14z^4 - \frac15z^5...

Substituting -z for z gives us

\log(1+z) = z - \frac12z^2 + \frac13z^3 - \frac14z^4 + \frac15z^5...

Adding the first and the third together and using

\log A - \log B = \log \frac AB

gives us:

\log\frac{1+z}{1-z} = 2( z + \frac13z^3 + \frac15z^5 + \frac17z^7...)

This is the basic series. However, it is not very useful for large values of z and even if you can compute log(x) for any x such that abs(z) < 1 it might still be useful to next consider how floating point numbers are represented in a computer.

Typically, floating point numbers are represented using three numbers: sign, mantissa and exponent. The sign is either +1 or -1 and is typically represented with one bit using 0 for +1 and 1 for -1.

mantissa is a number that represent the digits of the number. The exponent represent where the binary point (similar to the decimal point for decimal numbers) is placed.

Thus, the value of a floating point number is then:

value = sign \times mantissa \times base ^ exponent

The base is always a fixed constant and for most modern computers it is 2. Thus, to compute log(x) when x is of this form it is:

log(x) = log(sign) + log(mantissa) + exponent * log(base)

Here, the sign must be positive, the log function for real numbers is not defined for negative x, thus the sign is always +1 and log(1) = 0 so this contributes nothing.

What is left is the mantissa and the exponent. The exponent is known by simply extracting it from the number. So what we need to compute is log(mantissa). For most numbers the mantissa is always in the range 1 <= mantissa < 2 (or 1/2 <= mantissa < 1). This means that our z will actually always be 0 <= z < 1/3 (or -1/3 <= z < 0) and so the earlier mentioned series can be used to compute log(mantissa). log(x) is then simply log(mantissa) + exponent * log(base) and using 2 for base this is then easily computed.

log(2) can also computed using the above series as you get z = 1/3 or

\log 2 = \log \frac{ 1 + \frac13 }{ 1 - \frac13 } = 2(\frac13 + \frac13(\frac13)^3 + \frac15(\frac13)^5 + \frac17(\frac13)^7...)

Thus, if x is a positive floating point number with exponent n and mantissa m, then we can compute log(x) as follows:

log(x) = log(m2n) = logm + nlog2

From m, we compute z, as

\frac{1+z}{1-z} = m <=> 1 + z = m - mz <=> m - 1 = z(m+1) <=> z = \frac{m-1}{m+1}

From this we compute log(m) using z in the series above and then compute log(x).

For complex arguments z = x + yi it is probably best to convert x and y to polar co-ordinates and then use the identity:

z = x + yi = reθi + k2π

so log(z) = log(r) + θi + k2π

For complex arguments, the log function becomes multivalued in that any integer value for k gives a valid value since e2πi = 1.

The values r and θ can be found as: r = |z| = \sqrt{x^2 + y^2} and θ = arctan2(y,x) where y is the imaginary part and x is the real part of the complex number z.

I will move this to the main page 3 days from now if that is ok with everyone.

salte 10:11, 4 December 2006 (UTC)

[edit] Complex numbers and other domains

The section above indicates that complex logs will be added to the article, which is good. As it stands it's not clear what the domain of the logs, is though positive Real is assumed. Other domains, such as Complex are possible. I think there may be other possibilities too, such as for modular arithmetic, and groups e.g of matrices. David Martland 12:56, 7 December 2006 (UTC)

[edit] Added more on how to compute logarithm in computers.

Ok, I did what I threatened to do earlier and added in some stuff on computing logarithm in computers. Also added some information on how to obtain the series for logarithm. It is sketchy though but I think that is ok as this page is about logarithm and not on inverse function or integration of a series.

salte 12:53, 8 December 2006 (UTC)

[edit] Logarithms are not the only isomorphisms between R and R+

I've just removed, twice, the following (highlighted) statement:

For each positive b not equal to 1, the function logb (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms.

That's incorrect (and an exam question I've seen at least twice). The trick is that instead of considering the groups just as abelian groups, we can equivalently (proof left as exercise) consider them as Q-vector spaces, and as a Q-vector space, the reals have many automorphisms — in fact, more than there are real numbers, so if there is one isomorphism between R and another group, there are more than there are real numbers, so they can't all be of the form logb.

RandomP 21:33, 23 December 2006 (UTC)

Well, I put it back again before reading your explanation, and I must confess I have no idea what you're talking about. Can you provide either an example or a condition that would make the statement true, so that we can say something useful in place of what you're telling us is wrong? Dicklyon 21:51, 23 December 2006 (UTC)
What it's talking about is that in standard mathematics, you can define a bijective function f from the positive reals to the real numbers such that f(a*b) = f(a) + f(b) for all a, b positive real, but which isn't otherwise like a logarithm function. Unfortunately (from the pedagogic point of view) it isn't possible to really explicitly construct such a function, as far as I know; read axiom of choice for more information.
The condition you're looking for might be continuity or monotonicity; both make the statement correct. I'm not sure either statement is particularly useful.
RandomP 22:31, 23 December 2006 (UTC)

[edit] Jost Bürgi

I believe it violates NPOV to state that Jost Bürgi invented logarithms "first". Since he did not publish until four years *after* Napier, by which time Napier's work was widely known, de facto Bürgi does not deserve credit for being "first". In addition, it is not clear that Bürgi was first in any sense, since Napier worked on his version for 20 years before publishing. Is there documentary evidence that Bürgi started before 1594?

The neutral point of view is that Bürgi *independently* developed logarithms.

I have edited both the logarithm and Jost Bürgi Wiki articles to reflect this, giving Bürgi credit for independently inventing logarthms, but not for being first.

Nwbeeson 18:28, 4 January 2007 (UTC)

[edit] Function vs Number

I think that the logarithm can either be considered a function that yields a number or the number itself, depending on the convention. Perhaps it is best to include that in the lead paragraph? (See [5] and [6] where it is defined as a function; some other sites use "logarithmic function" instead of logarithm function though). —Mets501 (talk) 19:31, 26 January 2007 (UTC)

For people who understand numbers and functions, it's fine as it is; the lead is clearly desribing a function. However, for people who don't see that, perhaps you're right that we need to work the word function into it explicitly. But not carelessly, like your second ref that says "Logarithms are one of the functions...". And we might need to decide whether to describe one function of x and b, or an infinite number of functions of x, parameterized by b. We should perhaps look for a better way to put it. Dicklyon 22:15, 26 January 2007 (UTC)

Ah, I just noticed the point of your comment being Michael Hardy's removal of your contribution, which was this lead:

In mathematics, the logarithm is the function that is defined as the inverse of the exponential function. The logarithm of a number x in base b is the number n such that x = bn, ...

which he replaced with edit comment: (The logartithm of a specified number to a specified base is NOT a function; it is a number. A logarithmic function is a function. I'll be back later to add more.) with this older version:

In mathematics, the logarithm of a number x in base b is the number n such that x = bn,

I agree with your point that the log is both, and there's no conflict in your longer lead. Michael's comment is true, but I don't see how it applies, since you did not write that a the log "of a specified number to a specified base" is a function, but that "the log is the function that is defined s the inverse...". Sounds right to me. The second sentence refers to "logarithm of a number x in base b" which means, to anyone who knows about functions and numbers, the number the results from applying the function. Just like you could define square root as the inverse function of squaring, and then the square root of x is a number.

Is there a clearer way to put the fact that the term logarithm can refer both to the function generally and to the number that you get get by applying the function? Michael, ideas? Dicklyon 22:29, 26 January 2007 (UTC)

I think the explanation that I put at the beginning should come first, and the "function" idea later. Michael Hardy 00:50, 28 January 2007 (UTC)
Basically exactly what I was going to write before I got an edit conflict :-). Let's find out what Michael thinks... —Mets501 (talk) 22:33, 26 January 2007 (UTC)

Right after the very short introductory paragraph, I've added a new section titled Logarithmic functions to treat these issues. Michael Hardy 23:07, 26 January 2007 (UTC)

Micheal, from your comment and your edits, it seems you are disagreeing with us, and denying that the logarithm is a function or that "the logarithm function" is what it is called. And I disagree with your interpretation of "logarithmic function". A logarithmic function is any function that has the shape of a logarithm function. For example, in a MOSFET in weak inversion, the voltage from gate to source is a logarithmic function of the source-drain current. But it can NOT be written as just a log base b of I, because it needs a dimensional scale factor. The function is logarithmic, but it is not the logarithm function as defined in the article or as conventionally defined elsewhere. So, I think you've misinterpreted the words "logarithmic function" that you've read some place. Do you have a source for your interpretation? Dicklyon 23:24, 26 January 2007 (UTC)

What I wrote is entirely standard usage among mathematicians and---I think---others. I don't see that you shown that it conflicts with the example you've mentioned, involving dimensional quantities. And even so, the definition does not imply that there are no variations on the theme of the definition---of course there are; that always happens. "That I've read some place"?? I don't know how many books I've read about logarithmic functions in. Hundreds or thousands, probably. However, I will invite other Wikipedia mathematicians to look at this article with your points in mind, by mentioning it at Wikipedia talk:WikiProject Mathematics. Michael Hardy 23:52, 26 January 2007 (UTC)
Thanks, that kind of review will probably be useful. I'm not a mathematician per se, but to me "logarithmic" is a generic descriptor, while logarithm is the name of a function. I checked google book search, and found "logarithmic function" a bit more common than "logarithm function", but of course the latter doesn't begin to capture all the ways that people refer to the logarithm as a function. And of course if you've seen it in hundreds of books, finding one to share shouldn't be a burden. Dicklyon 00:28, 27 January 2007 (UTC)
Here, I'll respond for Michael. He's had a trying day. I've got a few books to share, Dick. My copy of Webster's Third New International Dictionary says that a logarithm is "the exponent that indicates the power to which a number must be raised to produce a given number <if B2 = N, then 2 is the logarithm of N (to the base B)>". My old (1976) copy of Encyclopædia Brittanica says "logarithm, the exponent or power to which a base must be raised to yield a given number. In the expression bx = N, for example, if b is the base and equal to 10 and N a number, equal to 100, then x is equal to 2 and is said to be the logarithm of 100 to the base 10." So it appears that the common usage of the word logarithm is that a logarithm is a number (an exponent, or a power), just as Michael maintains.
My even older (17th edition, 1969) copy of CRC Standard Mathematical Tables contains a table headed "Seven place mantissas of common logarithms", and another entitled "Table of natural or Naperian logarithms". Of course, if the word logarithm meant "a function", these tables would be headed "Seven place mantissas of the common logarithm" and "Table of the natural or Naperian logarithm". But they are not so labeled, because a logarithm is not a function; it is a number. The tables contain logarithms (plural), and those logarithms are numbers. If logarithms are numbers, then a logarithm is a number.
The source of your confusion is obvious. Since mathematicians commonly write f(x) =ln(x), it is easy to conflate the notion of a function (f) with the abbreviation (ln) and then to conclude that ln = logarithm to the base e. In fact, in standard mathematical notation, the abbreviation (ln) stands for "that function which assigns to every positive real number x its logarithm to the base e". Since people don't like such lengthy locutions, they often resort to shorter informal modes of expression. One ought not misinterpret the meaning of an abbreviation.
In fine, I will defend the truth of the statement "a logarithm is a number" as a matter both of grammatical precision and mathematical exactitude. DavidCBryant 02:59, 27 January 2007 (UTC)
It is completely standard to use the phrase "the logarithm" to refer to the function. Two examples from Google books:
1. "The logarithm is the function log: (0,oo)->R such that for any x>0, exp(log x) = x" Mathematical Analysis and Applications: An Introduction By J V Deshpande
2. "The natural logarithm is defined, as in the real domain, as the inverse of the exponential function" Elements of the Theory of Functions By Konrad Knopp
Moreover, the phrase "branch of the logarithm" only makes sense if logarithm is considered a function, not a particular value.
In short, I think that "logarithm" is commonly used to refer to a function, and also commonly used to refer to a value of that function. You might find that unfortunate, but the article might as well just point out that both meanings are used in practice and move on. CMummert · talk 03:47, 27 January 2007 (UTC)
David and CM, it seems obvious to me that you're both right. Thanks for the citations. I'd just like to point out that no reliable source showing one usage contradicts the other usage, unless they do so explicitly. I haven't seen any such explicit contradiction, and so it makes sense for wikipedia to represent BOTH common usages. Right? Dicklyon 06:43, 27 January 2007 (UTC)
The common practice on WP is to just describe how the terms are commonly used in the real world. For example, see the first para of uncountable set. This is a better service for readers than choosing the "correct" terminology and shunning all the others, because readers will come here from many diferent backgrounds, and will likely be confused if the terminology here seems to contradict the terminology they have already seen. It would be nice if real-world terminology was standandardized, grammatically correct, and logical, but it often fails these criteria. CMummert · talk 13:45, 27 January 2007 (UTC)

I don't see exactly what this discussion is about. It relates to terminology somehow, but I don't see quite how. I would use the words the natural logarithm to refer to the inverse function e^x, which is a synonym for the natural logarithm function. But it hurts my ear to read logarithm function in general, with the word logarithm used as an adjective, so I would write the synonymous phrase logarithmic function for euphony. Maybe somebody can state exactly what the disagreement is about? CMummert · talk 01:40, 27 January 2007 (UTC)

Sure: basically, the disagreement is whether the word "logarithm" by itself refers to a function, value, or both. —Mets501 (talk) 01:44, 27 January 2007 (UTC)
It can be used to refer to either. The natural logarithm is the inverse of the exponential function [7]. The natural logarithm of e is 1. Like many classical subjects (including trigonometric functions as well), the terminology is nuanced. CMummert · talk 01:56, 27 January 2007 (UTC)
Exactly, that's what I was saying in response to Michael Hardy's edit summary (The logartithm of a specified number to a specified base is NOT a function; it is a number). —Mets501 (talk) 01:59, 27 January 2007 (UTC)
Let's wait to give everyone else a chance to respond before pressing the issue further. The main question is the nature of any disagreements about terminology. CMummert · talk 02:08, 27 January 2007 (UTC)
Here's me jumping in. I will begin with a little history, a little etymology, from a fun source. At http://members.aol.com/jeff570/l.html, we find that Napier first used "logarithmus" around 1614 for a number, and its English equivalent, "logarithm", occurs soon after. Contrast that with "logarithmic function", which does not appear for another 200 years.
Let's move on to a mathematical distinction, which I will illustrate with "polynomial". The expression "4x3−3x" is a polynomial, formally a member of the ring Z[x]. It is an algebraic object, like a complex number, which we can play with in various ways. One of those ways is evaluation, substituting a number for x to get a number for the expression. Using this we can define the "polynomial function" p:R→R, where p(t) equals the evaluation of the aforementioned polynomial at t. I claim that we need to be equally careful about distinguishing a "logarithm" from a "logarithmic function", at least sometimes.
Which brings us to Wikipedia. Mathematicians abuse notation, and some writers are painfully sloppy. Non-mathematicians are rarely sensitive to our distinctions, and use words to suit themselves. Our job is more to describe than to dictate, but still we wish to set a good and helpful example of careful mathematical usage. We have adequate storage and bandwidth for two articles. So I would urge that "logarithm" be defined as a number, with a note that it may sometimes be shorthand for a "logarithmic function", especially the inverse of ex. --KSmrqT 17:27, 27 January 2007 (UTC)
Just to make things more complicated, in the (rather common) phrase "when logarithms were discovered", what was discovered wasn't really a function (and most definitely not a family of functions - other bases came later!), but the idea that the rather primitive integer-valued logarithm function defined on powers of 10 could be extended to a real-valued logarithm function defined for all positive real numbers.
I suspect that the hypothetical reader arriving at this article after having read that phrase would be rather disappointed, particularly if they are young enough not to have practiced long division extensively, and thus less likely to appreciate the significance of replacing it by table lookups and subtraction.
RandomP 02:19, 27 January 2007 (UTC)

It has been over a year since Jan. 20 2006 when this diff got rid of the log being a function and made it a number. For most of 2006, however, it was an "operation"; is that even sensible? Dicklyon 02:56, 27 January 2007 (UTC)

[edit] Context!!

OK, I thought this was obvious, but several people are not getting it:

This is context-dependent. In some contexts, it is incorrect to say the logarithm is a function; in other contexts, it is correct. I was pointing out that in one particular context, it is incorrect. I wrote:

"The logartithm of a specified number to a specified base is NOT a function; it is a number. A logarithmic function is a function."

So a bunch of people come along and point out, correctly, that in some contexts it is correct to regard the logarithm as a function. So far, so good. But then they go on to say that that proves I'm wrong. Michael Hardy 00:45, 28 January 2007 (UTC)

As you know, you're not wrong there. I don't see any actual disagreement in the discussion here about what is going on. It would be better to for those interested to just work on the article and let anyone who has an issue with what it says raise that issue here. CMummert · talk 01:21, 28 January 2007 (UTC)
Michael, if you'll review the comments, you'll see that I agreed you were right in that comment. I just didn't see how the comment supported the edit. I don't find the word "wrong" in the discussion, so you leave me wondering what you're referring to. Was it that I had a broader interpretation of the phrase "logarithmic function" than you had, as including more than the logarithm function? Dicklyon 01:59, 28 January 2007 (UTC)

[edit] The logarithm function

Looks like Riemann thought "logarithm" was the name of a function: Elements of the Theory of Functions of a Complex Variable: With Especial Reference to the Methods... by Heinrich Durège, 1896, says "We designate, after Riemann, by the name logarithm a function f(z), which has the property that..." google book. Dicklyon 07:02, 27 January 2007 (UTC)

You're missing the point! See my "context!!" comments above. Michael Hardy 00:47, 28 January 2007 (UTC)
Well, Riemann wrote most of his papers in German, and some in French, so what he probably thought of was eine logarithmus (German was his native tongue). Besides, the title of the chapter to which you linked is "Logarithmic and Exponential Functions", leading me to think that Durège (or maybe his editor) was prim and proper part of the time, and not so careful at other times.
Maintaining consistency in Wikipedia is tilting at windmills. I'm sort of a stickler for grammar, though, and I try to think clearly. A logarithm is a number. A logarithmic function assigns a logarithm to each positive real number. The phrase "logarithm function" is an abuse of the attributive noun. That being said, I really don't care how this article misuses the English language. I do see some glaring inconsistencies, though, which probably ought to be rectified.
  • The introduction suggests that logarithms can be calculated over a complex base. In practice this is almost never done; mathematicians invariably use the base e in complex analysis. The section "The logarithm as a function" states that the base b "must be positive and must differ from 1", and this is inconsistent with the "roots of unity" language in the introduction.
  • The caption under the illustration is worded very poorly. In particular, the locution "Logarithms of all bases pass through the point (1, 0) ..." suggests that logarithms are functions, and not numbers; this is inconsistent with the introductory paragraph.
  • Perhaps a reasonable compromise is to use phraseology such as "Strictly speaking, a logarithm is a number ... Informally, the word 'logarithm' is often used as shorthand for 'logarithmic function' ..." DavidCBryant 14:38, 27 January 2007 (UTC)
We say "the logarithm of five is 0.69897" (or 1.6094 or whatever, depending on the base). With respect to 5, "logarithm" is a function. With respect to 0.69897, "logarithm" is the value of that function, a number. It's no different from saying the "cosine is 0.3765" in one case, and "0.5 < cos x < 0.75" or "the cosine of x is between ½ and ¾" in another. Gene Nygaard 15:11, 27 January 2007 (UTC)
Exactly. Anything we say that would demote the function to be a secondary or informal meaning is unacceptable. The log is a function. The log of a number is a number. Just like cosine, square root, exponential, etc. Dicklyon 17:01, 27 January 2007 (UTC)
OK, fine. The physicists are going to dictate terminology to the mathematicians. It's OK by me. There are very deep mathematical reasons for distinguishing between numbers and relations (including functions). But, as I've already stated, maintaining consistency in Wikipedia is tilting at windmills. Don Quijote I am not.
This discussion is starting to remind me of QM 102, with Rochus von Vogt. Herr von Vogt would scribble some big quadruple integral over all space and time up on the blackboard, and ask if there were any questions. I'd say something like "Ought we not demonstrate that the proper integral converges on every bounded interval before asserting the existence of the improper integral?" And he'd throw chalk at me. Every time.
I still got an "A" in QM 102, because Vogt liked my paper about the positron. I was rather happy about including my interview with Carl Anderson in the paper, so I asked Herr von Vogt what he thought of that. He said, "You got an 'A' because you woke him up!" (Anderson was about 65 years old at the time, and spent most of the day sound asleep in his office.)
I still think that Wikipedia articles about mathematical subjects should strive for precision of expression. But I'm not going to make a big deal of it. I understand enough about physicists to know that in general, they just don't care about logical consistency in mathematics. So a number can be a function "by consensus", and it really doesn't matter to me. DavidCBryant 20:18, 27 January 2007 (UTC)
OK, but at Caltech they've got Math and Physics in the same division, so why the argument? I got my Caltech degree in Engineering, and skipped QM after Phys 2, so I usually avoid that level of argument. But I'm interested in trying to understand what you're saying here. Are you saying that in strict mathematical language there is not a function called the logarithm? And why? Here's another old math book that disgrees: Functions of a Complex Variable, y Edgar Jerome Townsend, 1915 Dicklyon 22:15, 27 January 2007 (UTC)
I'm copying down my comments from above, so someone will read them. First, history. At http://members.aol.com/jeff570/l.html, we find that Napier first used "logarithmus" around 1614 for a number; its English equivalent, "logarithm", occurs soon after. Contrast that with "logarithmic function", which does not appear for another 200 years.
Let's move on to a mathematical distinction, which I will illustrate with "polynomial". The expression "4x3−3x" is a polynomial, formally a member of the ring Z[x]. It is an algebraic object, like a complex number, which we can play with in various ways. One of those ways is evaluation, substituting a number for x to get a number for the expression. Using this we can define the "polynomial function" p:R→R, where p(t) equals the evaluation of the aforementioned polynomial at t. I claim that we need to be equally careful about distinguishing a "logarithm" from a "logarithmic function", at least sometimes.
And I will add a non-mathematical example, for those who still don't get it. A "telephone" is not the same thing as "calling on the telephone", even though we may say "Please telephone me", meaning the same thing as "Please call me on the telephone." Please do not bother to protest that no one will confuse the two; people learning new mathematics routinely get confused by far less! Nor are we saying that no one does, nor should, say "telephone me." Got it? --KSmrqT 02:57, 28 January 2007 (UTC)
I didn't respond the first time, but I thought your analogy with the polynomial and polynomial function was slightly strained, since the polynomial is not a number; the cosine versus cosine function that someone mentioned is a closer analogy. Would you argue that in that case a cosine is a number and that the cosine function should not be called simply the cosine? Obviously cosine, logs, polynomials, etc. are widely used as functions, and as far as I can tell it is perfect normal (possibly not perfectly rigorous) to refer to those functions simply as the cosine, the log, etc. In other words, I don't think we have any difference of opinion or understanding about that distinction, just about what is considered normal terminology. When logarithms as numbers were invented we didn't have a well developed math of real analysis, etc., so there was little chance that the logarithm would have been conceptualized as a function at that time. But certainly it is by the nineteenth century; I've found and mentioned several books that say the logarithm is a function, without agonizing over it. I never had any great cognitive dissonance with a table of logs or a table of sines being in conflict with thinking of it as a table of values of the sine or values of the log for different arguments, which is of course how it's laid out.
So I remain a little unclear on this fundamental question: do some of you claim that it is NOT usual and customary to say "the logarithm is a function" or "a logarithm is a function"? Dicklyon 04:03, 28 January 2007 (UTC)
This is belaboring the obvious, but I'll try one more time. Grammatically, "logarithm" is a noun. A logarithm is a number, and one ought to say "logarithmic function". Consider the analogous case of "exponent". "Exponent" is a noun, an exponent is a number, and we speak of the "exponential function". Nobody says "exponent function". At least, I've never heard it.
"That's a great deal to make one word mean," Alice said in a thoughtful tone.
"When I make a word do a lot of work like that," said Humpty Dumpty, "I always pay it extra." ;^> DavidCBryant 14:09, 29 January 2007 (UTC)
Nobody says exponent function because they are called power functions, still with an attributive noun. CMummert · talk 14:28, 29 January 2007 (UTC)
I understand that the Germanification of English continues apace. I can even parse the difference between "networkcomputersystemmaintenanceprogrammer" and "networkcomputersystemdevelopmentprogrammer" without a dictionary. That doesn't mean I have to like it – or write that way, either. DavidCBryant 16:43, 29 January 2007 (UTC)

[edit] Common usage

Rather than argue about the logic of how we think terms should be used, shouldn't we be following reliable sources and representing actual usage there? For example, to get a quick survey of sources more reliable than web pages, take a look at google book search. Search for phrases "derivative of the logarithm", "derivative of the logarithm function" and "derivative of the logarithmic function", and it becomes apparent that that former is an order of magnitude more common than the other two. It seems very clear that the word "logarithm" is used to refer to the function, and for us to assert reasons to avoid that seems un-wikipedia-like. No? Dicklyon 17:32, 29 January 2007 (UTC)

The article currently says

The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

I don't think there is any actual disagreement over the content of the article. The discussion on the talk page seems to be about grammar, usage in the real world, etc. but the comments don't seem to directly relate to any part of the article. Can anyone describe any specific changes they would like to see in the article text? CMummert · talk 17:45, 29 January 2007 (UTC)


[edit] Article mentioned in Economist's website "top story"

Just an FYI, but the top story on The Economist's website calls this article "admirable" and cites it as an example of a quality Wikipedia page. Not sure there are any special actions/notice needed... --Dfred 01:55, 11 March 2007 (UTC)

[edit] Different image (svg)

I've just uploaded a new image of the logarithmic functions (adding log0.5) from a german image used in the norwegian article. I created it in gnuplot, and it is a 640x480px SVG-image. SVG has the advantage that it is scaleable, and does not work the same way as bitmap-images. You can scale it as much as you want to, without a loss in quality. I have not altered the article, because the one that is now used, looks better (but has poor bitmap quality). The image is located here: Image:Logarithmic_functions.svg. I altered the norwegian article, located here (image in 300px): no:Logaritme. Chrtsta 21:30, 16 March 2007 (UTC)

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