Talk:Logarithm

Link in Computer Science section

Does a major part of the whole sentence need to be linked to the change of base identity in this section? I've edited it so only the 'following identity' text is linked; can someone please change this back if the original was better? Thanks. Dakuton (talk) 21:19, 28 October 2009 (UTC)

Tables of Logarithms

I think this section should be moved to a separate page and expanded. Mathematicians who have been educated in the last 30 years have never seen a table of logarithms because they have been replaced by electronic calculators. The history of the availability of tables of various number of decimal places should be traced. The historical effects of their invention and use in astronomy cannot be overemphasized.

Trojancowboy (talk) 19:47, 5 February 2010 (UTC)

Difference of root and reciprocal

I was bored one day, I was playing around with the ln(x) function on a piece of graph paper and a calculator. :-P I noticed an unusual property: $Log(x)=\lim _{n\to \infty }n{\frac {x^{1/n}-x^{-1/n}}{2}}$ I've checked many different places and verified it. I can't find anyone else mentioning this property, either. Should it be included on Wikipedia? I think it probably should be, but I can't find a good place to slip it into the article... Timeroot (talk) 05:43, 14 February 2010 (UTC)

Let's see - for a fixed value of x:

$x^{1/n}=e^{\ln(x)/n}=1+{\frac {\ln(x)}{n}}+O(1/n^{2})$

$x^{-1/n}=e^{-\ln(x)/n}=1-{\frac {\ln(x)}{n}}+O(1/n^{2})$

$\Rightarrow n{\frac {x^{1/n}-x^{-1/n}}{2}}={\frac {n}{2}}\left({\frac {2\ln(x)}{n}}+O(1/n^{2})\right)=\ln(x)+O(1/n)$

$\Rightarrow \lim _{n\to \infty }n{\frac {x^{1/n}-x^{-1/n}}{2}}=\ln(x)$

To include this in the article, you would need to find a reliable source for this limit. Without a reliable source, it is original research. Gandalf61 (talk) 09:46, 14 February 2010 (UTC)

or publish it yourself and hope that someone else picks it up. —Preceding unsigned comment added by 83.233.83.192 (talk) 15:13, 30 October 2010 (UTC)

What's the name for the argument of the logarithmic function?

When reading the exponential term a^x, one can say "exponentiation - of a - to the exponent n". However, one can also use the explicit name "base" for a, and say: "exponentiation - of the base a - to the exponent n". My question is about whether one can also use any explicit name for x - when reading the logarithmic term log_ax, i.e. by saying something like: "logarithm - of the blablabla x - to the base a"...

HOOTmag (talk) 20:47, 15 February 2010 (UTC)

Change of Base section

I didn't find the change of base section clear enough to be fully understood. I have found a simpler proof here [1].(includes proofs for the other 3 laws of log) Mohamed Magdy, Thank You! (talk) 20:53, 25 February 2010 (UTC)

The "simpler proof" on that web page is actually the same proof that's given in the "change of base" section of this article. Michael Hardy (talk) 23:07, 25 February 2010 (UTC)

Properties of Logarithms

Some of the properties of Logs are not entirely correct.

Log(AB) does not only equal Log (A) + Log(B) (This is only a special case, when both A>0 and B>0).

The general property is Log(AB)= Log |A| + Log |B|. This is because both A and B can be less than 0.

Why is this not in the article?

Another property of Logs that is missing is the following:

Log to the base b^x (A^y)= (y/x)• Log to the base b (A) David Yakubov (talk) 05:11, 4 March 2010 (UTC)

"Why is this not in the article?" Probably because no one has incorporated it yet. As for your first point, how does this look to you:

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

$\log(xy)=\log |x|+\log |y|\,.$

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers' absolute value.

My understanding of the topic is meagre at best, and I see some potential for complication here, so I'm not incorporating this into the article myself – please correct or improve as necessary here or fix the article directly. Be bold! /ninly_(talk) 13:53, 4 March 2010 (UTC)

The wording could be tightened up but the way to do that is to make it clearer that this article in the main only deals with positive real numbers. No absolute values should be put in - the equations don't need to be changed. There is a small section on the logs of negative and complex numbers which points to another article. Dmcq (talk) 15:04, 4 March 2010 (UTC)

If x, y > 0 then

$\log x+\log y=\log(xy).\,$

That's the essential idea one should convey. One could mention the corollary in an appropriate context, but it shouldn't have anywhere near as much prominence as this main idea. Michael Hardy (talk) 16:06, 4 March 2010 (UTC)

A more fundamental answer would be: the formula $\log(xy)=\log |x|+\log |y|\,$ is not in the article because it is not generally true. It fails for example if x>0 and y<0. It is possible to extend the logarithm to the complex numbers (excluding 0), albeit it as a multivalued function. For that function $\log(xy)=\log x+\log y\,$ for all values of x and y unequal to zero (seen as multiset). For most values of x and y, the equation stated above (with the absolute values) is not true. −Woodstone (talk) 16:35, 4 March 2010 (UTC)

The numbers in discussion here are the real, and not any other set of numbers. Of course the property fails for x>0 and y<0, because for the reals, both x and y (or any even amount of variables) have to either all be positive or all negative, for the property to remain true.

Another important property not listed in the article is this one:

• ${\log _{a^{q}}{b}}^{p}={\frac {p}{q}}\log _{a}{b}$

List of Properties

• $\log _{a}(bc)\ =\log _{a}|b|+\log _{a}|c|\quad (bc>0)$

• $\log _{a}{\frac {b}{c}}=\log _{a}|b|-\log _{a}|c|\quad \left({\frac {b}{c}}>0\right)$

• $\log _{a}(b^{p})=p\ \log _{a}|b|\quad (b^{p}>0)$

• $\log _{a^{k}}b={\frac {1}{k}}\log _{a}b$

• ${\log _{a^{q}}{b}}^{p}={\frac {p}{q}}\log _{a}{b}$
• $\log _{a^{k}}b^{k}=\log _{a}b$
• $\log _{a}b={\frac {\log _{c}b}{\log _{c}a}}$

$\log _{c}a^{\log _{a}b}=\log _{c}b\Leftrightarrow \log _{a}b\cdot \log _{c}a=log_{c}b\Leftrightarrow \log _{a}b={\frac {\log _{c}b}{\log _{c}a}}$

• $\log _{a}b={\frac {1}{\log _{b}a}}$
• $a^{log_{c}d}=d^{log_{c}a}$

The Russian article on Logarithms has all the properties as well as many of their proofs.

David Yakubov (talk) 01:57, 5 March 2010 (UTC)

As stated before, the equation:

log ab = log a + log b is more generally true than

log ab = log |a| + log |b|

This more complex and less valid formula does certainly not belong in the lead section. −Woodstone (talk) 07:11, 10 March 2010 (UTC)

...and all the other properties there are easily derived from three basic ones (change of base, exponent->coeff, coeff->addend), and still many of them only work once restricted to positive/integer numbers only. —Preceding unsigned comment added by Timeroot (talk • contribs) 23:23, 24 March 2010 (UTC)

apropos history section

Please re-read the section which you re-inserted - it talks about the etymology of the term "algorithm" and also has justification being placed in an article about the history of algebra. It does not fit here.
The term "logarithm" has other etymological roots.
217.236.174.10 (talk) 15:35, 10 April 2010 (UTC)
PS Of course we could copy-paste the history section of the mathematics article - but it would not make much sense, would it? 217.236.174.10 (talk) 15:40, 10 April 2010 (UTC)

Information & complexity

"with k bits (each a 0 or a 1) one can represent 2^k distinct values, so any natural number N can be represented in no more than (log_2 N) + 1 bits."

Natural number is an ambiguous term. According to that article, it may or may not include 0. While "no more than" covers both cases, wouldn't it be clearer to chose one definition, either positive integers or non-negative integers? Thus:

"so any positive integer N needs (log_2 N) bits to represent it."

or

"so any non-negative integer N needs (log_2 N) + 1 bits to represent it."

Dependent Variable (talk) 07:53, 12 May 2010 (UTC)

Go ahead! (I would prefer the first of the two). Jakob.scholbach (talk) 11:47, 12 May 2010 (UTC)

However, how do you represent N=1 in zero bits? Perhaps you mean to say that log₂ N bits are needed to encode all integers from 1 to N (inclusive)? Then (1) carries no information, 0 bits, for (1,2) you need 1 bit, for (1,2,3) 2 bits. You should indicate the rounding up of the value explicitly. −Woodstone (talk) 12:16, 12 May 2010 (UTC)

It could be phrased a bit better. The number N can be represented using 0 bits if we know in advance that's the only number that could be used in a context. For instance the answer to how many side has an octahedron needs zero bits to represent it. What's meant is that any number in the range 0 to N can be represented using that number of bits. Dmcq (talk) 13:13, 12 May 2010 (UTC)

That does not address all concerns. To encode N items, the number of bits needed is log₂ N rounded up to the nearest integer. −Woodstone (talk) 05:58, 14 May 2010 (UTC)

Sorry, I'm new to the wiki, I edit the article in a way adresses all concerns but as I couldn't believe the situation before I edit, I started to investigate the history. Both my edits about base (unnecessary and inconsistent concerns about "zero") and guaranteeing an integer value was in the history. I know these are elementary stuff. So maybe there was an argument about it and I missed? —Preceding unsigned comment added by Oz an (talk • contribs) 01:41, 7 July 2010 (UTC)

Antilogarithm

I refer to the statement within the article:

"At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers they represent, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and,

while never used extensively in mathematics, persisted in collections of tables until they fell into disuse."

This is just not true. To obtain a mathematical solution using a Log table the Antilog has to be used to provide the end result from the logs. So Antilogs are an interegal part of using Logs. Thus they can never not be used.

In fact even today there are no spreadsheets programs that provide both an @ function for Logs and an @ function for Antilogs. Consequentally one is left with having to use printed log tables to produce the Antilog solution, since the production of logs is useless without the antilog. —Preceding unsigned comment added by P. Dickins (talk • contribs) 09:58, 7 July 2010 (UTC)

I don't know what you mean by an @ function. You weren't looking for 'antilog' were you?, you just use 10^x or exp(x) for that. With the printed logs it is perfectly easy to do ones work just using the log table to do antilogs. Dmcq (talk) 10:16, 7 July 2010 (UTC)

General logarithmic function

By looking at the article, one may notice that it follows the traditional way of defining a logarithm and like so many texts, it does not include the definition of the exponent being irrational. So many texts just assume that the base to a power is defined for the base greater than zero and the exponent is any real number and the usual laws of exponents are valid that it looks the Wiki article just wants to follow suit. If so, fine. However, there is a general formula that applies to any exponent, real number (rational or irrational) and to any base. Neither the Russian nor the German page has it. Perhaps the French...JDPhD (talk) 21:44, 19 July 2010 (UTC)

The majority of textbooks dealing with logarithms are introducing the subject to students so they concentrate on a simple beginning, namely rational exponents. Wikipedia's coverage should also include a rigorous coverage of irrational exponents. If you have a suitable source that you can cite in the article please go ahead and add that rigorous coverage. Dolphin (t) 22:29, 19 July 2010 (UTC)

The new "general" section had no contents, just a reference to an inaccessible book, so I took it out. If there's something that should go there, put it in. It would be nice if there's an online source, too. Dicklyon (talk) 02:50, 22 July 2010 (UTC)

How about moving section logarithm#Alternative definition via integrals up as general definition? −Woodstone (talk) 03:50, 22 July 2010 (UTC)

Why? The beginning of the article is accessible to schoolchildren without calculus, this would make it inaccessible. This is an encyclopaedia not a calculus textbook. Dmcq (talk) 06:58, 22 July 2010 (UTC)

Also, the Alternative definition via integrals is not the general definition. One thing more, when I began this section that was erased, I was going to include a "hidden comment" asking the editors not to erase it until it's done. For one thing, there will be two sections with two drawings and the references but the whole thing cannot be done in a jiffy. The book that Dicklyon mentions has ISBN for the new editions; the one cited in the article happens to be the edition of 1959. So if the rest of you feel ok with a "hidden comment" (not to erase anything until it's done) I'll go right ahead and start. I promise you, you won't be sorry. JDPhD (talk) 19:14, 22 July 2010 (UTC)

It's probably best to prepare something like that first in a user sandbox until you have enough to look reasonable if its going to take some time. If you want to develop in an article putting in the citations and basic discussion first and doing the tidying up of the language later is less likely to get removed. Dmcq (talk) 20:49, 22 July 2010 (UTC)

I agree that using a personal sandbox is the best way to develop extensive changes to any article, especially as no-one will interfere with your changes while they are work-in-progress on your sandbox. Information about creating a personal sandbox is available at WP:SP and WP:USERSUBPAGE. Dolphin (t) 23:14, 22 July 2010 (UTC)

By "inaccessible" I meant I can't find access to the book contents online, so can't see what you're getting at and try to help. Surely there are other books, accessible on amazon or google, that could be used as source for what you're trying to do, yes? Dicklyon (talk) 02:46, 23 July 2010 (UTC)

As alternative to sandbox, you could post your proposed new section here on the talk page, and invite help to get it into good enough shape to put into the article. Nothing is lost, by the way; just use the history to go back to your version, or the corrected one after it, and edit to copy out the section source that you want to reuse. Dicklyon (talk) 02:49, 23 July 2010 (UTC)

OK let's try it out first, here in the Discussion page. Of course, help is always welcome.JDPhD (talk) 19:46, 23 July 2010 (UTC)

The general expression of the logarithm

So far in this article, the function $y=a^{x}$ has been defined only for rational values of "x", except in the particular case when $a=e$. We shall now consider the case in which "a" is any positive number.JDPhD (talk) 20:13, 23 July 2010 (UTC)

There's the following obstacle to continuing. From now on, in the proposed addition, everything except the two diagrams is original research. It is rather difficult to find references to match the original research. Therefore, it does not satisfy Wikipedia standards. So, let's fold the page here and desist from the enterprise.JDPhD (talk) 18:58, 26 July 2010 (UTC)

That sounds like the right decision. Dicklyon (talk) 00:26, 27 July 2010 (UTC)

Simplification

This recent edit "simplified" the proof that the real log is well-defined. Actually I just wrote this elongated explanation in order to make it accessible to people who don't know about (inverse) functions etc. I'm going to revert that simplification unless somebody convinces me of the contrary. Jakob.scholbach (talk) 13:42, 1 August 2010 (UTC)

Thinking about it again, I decided to revert this for the moment. Jakob.scholbach (talk) 13:51, 1 August 2010 (UTC)

I simplified your explanation because I don't see the point of this lengthy discourse on a special case of a more general result. The general result - a continuous strictly monotonic real-valued function has an inverse - is both intuitively obvious and simple to prove rigorously. The fact that the exponential function has an inverse for b ≠ 1 is not surprising, and not worth writing paragraphs about. Wikipedia is not a textbook - "The purpose of Wikipedia is to present facts, not to teach subject matter". Gandalf61 (talk) 14:21, 1 August 2010 (UTC)

I'm aware of wp:nottextbook and I agree my draft so far is somewhat borderline. The problem with your approach is that it is practically unintelligible for a 15, 16 year high-school reader, which is the audience we should care about at this point. I tried to avoid the words "function", "continuous", "monotonic", inverse function, since these words will not be known to this audience. Right? So the question is: how can we give a meaningful explanation, which is accessible to a broad audience, yet stay simple and concise? How about a little animation depicting the situation (think of a graph and repeating this bisection process a few times)? Jakob.scholbach (talk) 14:48, 1 August 2010 (UTC)

A minor WP:MOS thing: "we" is unencyclopedic language. Mathematicians (including myself) have certainly a reflex of talking to the reader... Jakob.scholbach (talk) 14:48, 1 August 2010 (UTC)

Your explanation is neither simple nor concise nor complete. For example, you do not explain why your bisection process converges to a limit - you are implicitly using the completeness of the reals here. Any reader bright enough to wonder whether the exponential function really has an inverse is going to wonder about this too. A simple and concise explanation will use the correct and accurate terminology, not baby words like "bigger and bigger" and "closer and closer". The interested reader can look up the linked articles, and will then have learnt something. I have restored my simplified version, modified to avoid the use of the second person. Gandalf61 (talk) 15:12, 1 August 2010 (UTC)

I am the first to agree that my draft is only a humble start (which I hope to improve). Let's, for now, not worry too much about my admittedly childish wording.

The more fundamental question is this: for whom do we want to write this article here? You didn't seem to notice that in the next subsection there is already pretty much what you call the "simplified" explanation using calculus language. (Btw, your version is inaccurate as the strict monotonicity is not enough, the unboundedness to the right and boundedness by 0 to the left is needed). So, for the bright (I would rather say, educated) reader we have this. However, this is one of the 500 most viewed math articles, and >95% of our readers won't know what continuous means. Linking to continuous function, say, directs them to a mess(!), and I dare say nobody who does not already know this notion will dig trough it and come back and apply it to logarithm. Do you agree with that? Jakob.scholbach (talk) 15:42, 1 August 2010 (UTC)

About simplicity: specifically in what way is the more detailed approach not simple? Or the other way round, in what sense is your edit a simplification? (As opposed to replacing, after all, relatively easy facts by more fancy terminology?)

About completeness: I agree that the explanation I put is not 100% waterproof in the standards of a (under)graduate level. But certainly, the completeness of the reals is about the last thing I would care about in the presentation here. Most, say, engineering textbooks would not mention that. The alternative "is a continuous, strictly monotonic function from the reals to the positive reals, so it has an inverse function from the positive reals to the reals" may be more "complete" (granting that the single word so just encapsulates all of these things), but is without any doubt much less understandable. Jakob.scholbach (talk) 15:42, 1 August 2010 (UTC)

I drafted animation illustrating the proof (lots of details have to be improved). Any thoughts about that? Jakob.scholbach (talk) 16:51, 1 August 2010 (UTC)

I must generally agree with Gandalf here, the explanation is actually looking like a week case of WP:OR: an uncited proof. It show the problems of such with a lack of mathematical rigor, and an unconventional treatment of the material, Spivac for example defines the log of a real number through the evaluation of an integral. I do feel this section would be better treated by referring to the appropriate theorems. I'm not convinced by the need for a layman's explanation, we are considerably far down the article in a section about Analytic properties so I would expect some level of mathematical soptication by this level, many readers will be happy with just a statement that it can be defined to cover positive reals, by using techniques of analysis.--Salix (talk): 17:50, 1 August 2010 (UTC)

Log functions graph

Nice article :) Just one question in passing: why base 1.7 in the log functions graph? It seems an unusual choice of number - is this purely to give visual symmetry around the ln curve or does it have some interesting mathematical property? EyeSerene^talk 13:06, 2 August 2010 (UTC)

I believe 1.7 was chosen purely to provide visual symmetry. It needed to be a number between e (2.718...) and unity (one). The number 1.7 is not the arithmetic mean of 2.718 and 1.000, but it does provide good visual symmetry. Dolphin (t) 13:39, 2 August 2010 (UTC)

Yes, it does. The only thing I could think of (numerically) was that the dB scale is a log scale, 3 is a doubling of power/intensity on this scale, and 1.7 is close to √3... but that did seem a bit tenuous :) EyeSerene^talk 13:45, 2 August 2010 (UTC)

History of the logarithm

I find with certain surprise that it was not Napier who discovered logarithms, but an unknown indian mathematician in the 8th century!! Although the sources clearly state it, I have my doubts. Indian writers are known for their nationalistic bias, driven by the need to boost indian national pride. In an attempt to counterbalance what they see as "Eurocentrism", they created another ethnocentrism called "Indocentrism". Just look at the various claims that 14th indians invented calculus 300 before Newton and Leibniz, just because some calculus related ideas are found in thei works (forget Archimedes and the other greeks who did similar works 1500 years before). Other known cases include the discovery of heliocentrism by astronomer Ayharbata, and the determination of the speed of light by Sayana, a commentator of the Vedas (don´t ask me how he did it, not even Subhash Kak knows).

So far, I have only seen this claim in Jain run websites, (which I don´t think are very reliable) and the works of R.C and A.N. Singh and Gupta. No other scholar working in the field of indian matehmatics ever mentions that they worked with logarithms before Napier. Can we ignore 100 works who don´t mention this, and listen to these 2? --Knight1993 (talk) 20:55, 11 August 2010 (UTC)

I've looked into the source and there is good evidence that they did study the process of "how many times can you halve a number". The A.N Singh references is basically a direct translation of the original text. You do need to be very careful with the claims, they did not invent our modern conception of logarithms, from the text it only seems that it was applied to integers, and chiefly as a way of looking at very large numbers. No evidence that it was used as a easy way to carry out multiplications, or many of the other applications.

The evidence for the logarithm precursor is much stronger than the speed of light claim, there are several formula for properties of the logarithm precursor spanning several . For the speed of light it just seems to be one sentence with a number which happens to be the approximate value.

So Napier is the discoverer of logarithms, but an Indian did look at something similar before hand. --Salix (talk): 22:21, 11 August 2010 (UTC)

M Stifel did a similar thing, in that he made a sequence of integers aligned with a sequence of powers of two. But if you look at the refs I added, you'll see that historians don't really regard that as a discovery of logarithms. There may be lots of other things in exponential or logarithmic relationship that others have commented on, but I don't think we can add all these as discoveries of logarithms. Dicklyon (talk) 04:02, 12 August 2010 (UTC)

Date of log x (base e) to log x (base 10) and ln x (base e) notation switch?

I've always been under the impression that log x, implied base 10, and ln x, implied base e; but as I have come to find, in the year 1900, log x implied base e. Does anyone know in what year the official switch occurred? I'm trying to dig out the history of how the notation for the Boltzmann entropy formula switched from S = k log W (1900) to S = k ln W (modern). --Libb Thims (talk) 10:49, 2 September 2010 (UTC)

I'm not sure there was a such a switch. There are (still, so to say) areas which write log for base e, as noted in the article. Jakob.scholbach (talk) 14:45, 2 September 2010 (UTC)

Different authors use different notation. I doubt there was ever an official position, or an international decision to switch from one notation to another. Dolphin (t) 22:46, 2 September 2010 (UTC)

There's no such thing as such an "official switch"; there are only conventional usages. There is no authority that issues decrees about such things. "log x" means the base-e logarithm of x when that notation is used in contexts where that is the appropriate base. That is commonplace usage among mathematicians today. In the present day, the "ln x is also used. Not so many years ago, I used to tell people that "ln x" is used only in textbooks for first- and second-year calculus, whereas adults write "log x" for natural logarithm of x. In Paul Halmos' autobiography, he ridiculed the "ln" notation then (in 1984)) appearing in those low-level textbooks, saying no mathematician had ever used it. That was a bit of an exaggeration at the time it was published, and is not true today. But "log x" continues today to be understood as logarithm to the appropriate base, where the appropriate base varies with the context. Michael Hardy (talk) 22:49, 2 September 2010 (UTC)

At Natural_logarithm#Notational_conventions we find this:

Mathematicians, statisticians, and some engineers generally understand either "log(x)" or "ln(x)" to mean log_e(x), i.e., the natural logarithm of x, and write "log₁₀(x)" if the base 10 logarithm of x is intended.

Some engineers, biologists, and some others generally write "ln(x)" (or occasionally "log_e(x)") when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x).

In most commonly-used programming languages, including C, C++, SAS, MATLAB, Fortran, and BASIC, "log" or "LOG" refers to the natural logarithm.

In hand-held calculators, the natural logarithm is denoted ln, whereas log is the base 10 logarithm.

In theoretical computer science, information theory, and cryptography "log(x)" generally means "log₂(x)" (although this is often written as lg(x) instead).

Michael Hardy (talk) 22:55, 2 September 2010 (UTC)

Ludwig Boltzmann tombstone, built in circa 1935.

In 1901, Max Planck used the following logarithm notation.

$S=k\log W\!$

This notation convention was engraved, in circa 1835, on Ludwig Boltzmann's tomb (adjacent), a tribute to his 1872 paper “Further Studies on the Thermal Equilibrium of Gas Molecules”, in which he is said to have introduced a variation of this equation (I still have yet to read this paper, being that no readily available English translations exist). It seems, however, that in the years 1930 to modern, the "ln" notation has come into popularity, e.g. 1996 book cover, Crease's 2004 list (see "Equation" section) of the 20-greatest equations ever, etc.,

$S=k\ln W\!$

I have always been a bit puzzled why these two different notations are used. On Wednesday, someone commented to me about this matter:

"The use of "log" refers to (and always did) the natural logarithm (base e) in this formula, and it was only afterwards that log was used to denote the base 10 logarithm and ln the base e."

I'm guessing that some modern writers have tended to switched to "ln" notation, so as to be absolutely clear that base e is the being used, where as "log" leaves the interpretation open to either base e or base 10. I'm sure there's a story as to how this switch occurred. --Libb Thims (talk) 11:39, 3 September 2010 (UTC)

However note that there is a linear relationship between logs in different bases. So the formulas $S=k\log _{e}W\!$ and $S=k\log _{10}W\!$ are equally valid, only they lead to a different numerical value of the constant k. −Woodstone (talk) 11:59, 3 September 2010 (UTC)

My computer is presently operating Microsoft Windows XP. It has a scientific calculator with one button labelled log and another button labelled ln. Dolphin (t) 04:00, 9 September 2010 (UTC)

Path to GA?

I'm hoping to take this article to GA status in the near future. To get a better standing, I'd like to solicit further input of people watching this. What do you think needs to be done before going to GA?

Myself, I see at least the following to-do-items:

• check and substantiate claims in history section
• explain Napier's version of logs more thorougly
• maybe explain discrete logs (a little bit) more. add Zech's logs there
• overhaul the "Calculation" section. What algorithms are used in, say, C++ or Maple for numerical calculations

Jakob.scholbach (talk) 08:50, 26 September 2010 (UTC)

• A minor comment: the complexity theory section jumps rather abruptly to a discussion of fractal dimension. This could use some more exposition. A separate section might even be warranted. 71.182.217.132 (talk) 22:05, 12 October 2010 (UTC)

Should be better now. Jakob.scholbach (talk) 21:31, 17 October 2010 (UTC)

• Another minor comment: the first sentence wikilinks to both power and exponent ("is the power or exponent to which the base must be raised"), but these go to the same articles. Is there any meaningful difference between power and exponent? This could raise a question in the readers' mind. I'm pretty sure these are the same and a synonym is just being used, but I think one word is enough - exponent. II | (t - c) 20:01, 17 October 2010 (UTC)

I think exponent is the more correct term, but power is a term that lay readers may have used for this. As I understand it, power is not really right; that is, in "the third power of 2 is 8", the "power" refers to the 8, and the exponent is 3. But it's not unusual to see "raise 2 to the power 3". Dicklyon (talk) 21:03, 17 October 2010 (UTC)

I opted for exponent. Jakob.scholbach (talk) 21:31, 17 October 2010 (UTC)

Formula formatting consistency - informal RFC

I think that the looks of the current article is in a pretty bad shape, so yesterday I fixed the first little section (containing two HTML-formatted standalone equations and two math-formatted ones) from this into this, with the intention to provide the entire article with a consistent formula formatting (see the edit summary).

As a result I got a talk page message from Jakob.scholbach (talk · contribs), which I think belongs on this article talk page, so I copied it here below, with a comment.

(copied from talk page)

I was willing to ignore my qualms about your and John's formatting doctrine over at talk:complex number. Now at logarithm, however, you seem to just override your own rule which I might paraphrase as: don't change an article's formatting without good reason. Even though this is a totally nitpickishy issue, you should still apply commonsense to it. Which means: simply because there are two math markup formulas and 5 HTML formulas nearby, this does not entitle you or anyone else to start changing the corresponding section. Then maybe you do the surrounding two sections, then, ooops, now we have an inconsistently formatted article, let's do whatever we want. I kindly, but firmly ask you to respect other people's work. I invested a lot in logarithms, which does not mean it is a perfect article, but which does mean that quite some thought has gone into this article, including markup. I'm happy to disclose my reason not to use math markup in elementary formulas which can be done using HTML: it just looks inconsistent, cause

$|z_{1}+z_{2}|=5$

will render for most users in this intermediate formate, not the same as

z₁ + z₂ = 5

(which is perfectly consistent with the text font in the main article) nor

$\displaystyle |z_{1}+z_{2}|=5$

which we (regret it or not) have to use for complicated, non-HTML-able formulas.

You were concerned about inconsistencies in complex numbers. I was unwilling to discuss it til the end there, but if you stir it up again, I have to ask you: don't introduce even more inconsistencies. If you have a liking for polish markup, there are plenty of articles which sorely need the work. Logarithm does not need it, IMO. Jakob.scholbach (talk) 07:38, 18 November 2010 (UTC)

(end of copy)

Jacob, I don't see any inconsistency in the fixed formatting of this first section.

The idea is not to have this:

$|z_{1}+z_{2}|=5$

but this

$|z_{1}+z_{2}|=5\,$

which is indeed the same as this

$\displaystyle |z_{1}+z_{2}|=5$

Note the added "\," at the end, to force PNG-rendering - see Help:Formula#Forced_PNG_rendering.

Anyway, this —including the relevant guideline— has been discussed in Talk:Complex_number#Brushing_over and Talk:Complex_number#Work_in_progress, so what do other contributors think about this? I also have asked JohnBlackburne to come over and comment as well. DVdm (talk) 10:35, 18 November 2010 (UTC)

I agree, that for both consistency's sake and presentation's sake TeX formatting should be used wherever possible. Consistency because it is jarring for the article to change from one format to another, sometimes within the same section, without good reason. Tex rendering uses a different font, is larger and uses spacing and relative sizes much more appropriate for math formulae. Even when it drops down into text rendering the font and size is much more consistent, and this is something users can override by e.g. forcing all TeX to be images, or even render TeX as SVG, though this is slow at least for me; see Help talk:Displaying a formula#Formulas as SVG?.

Presentation because HTML is not especially good for rendering maths: it is too small, difficult to tell some things apart and tends to run italic text into non-italic text and get other spacing wrong. This article is a prime example as there's text with subscripts of superscripts which are simply too small. Colour has been added to presumably get around this and make it easier to tell them apart, but it only makes it worse: colours, especially bright colours, make text less readable, don't work for colour-blind readers or those using monochrome displays. It's why after a few years of experimentation black text on a white background is used everywhere, and why we have a guideline, WP:COLOR.

In some way the guidelines are unclear on this as they do not mandate TeX rendering, but instead leave it up to editors. But looking elsewhere the guidelines are clear, especially the first of the general principles on internal_consistency, and except for short articles with only simple formulae consistency can only be achieved by using TeX. It is also noted at MOS:MATH#Very simple formulae that formatting can be changed to make the entire article consistent, again only possible using TeX.--JohnBlackburne^words_deeds 12:03, 18 November 2010 (UTC)

First an aside @DVdm: calling a specific editor, whose opinion you already know and agree with is called forum shopping and is frowned upon.

Second: the usage of colors on one occasion in this article may have been unfortunate. Even though the colors are certainly not light and I'm not convinced that these dark colors hinder the understanding of any color-blind reader, I appreciate any edits removing the colors and expressing in words what the colors would have said.

Now, for tex vs. non-tex. Thanks, John, for acknowledging that MOS:MATH does not prescribe the usage of math markup. The guideline you do cite does not mandate your point. I think you misinterpret or overinterpret it. If everything in an article should have the same style, then you would have to replace html markup in any mathematical expression, not only standalone equations. This is nowhere done, and in fact there is a guideline saying that mathematical expressions in the main text should not be texified where possible, to get a smoother lineage).

Accepting this, the only conclusion is, that there is not a single (except for stubs and non-math articles) math article which is entirely coherently formatted. The closest approximation to coherence is to use html wherever possible. I don't take your point about the tiny sub-superscripts. Look at this:

(b^log_b(x))^p = b^{p · log_b(x)}.

vs.

$(b^{\log _{b}(x)})^{p}=b^{p\cdot \log _{b}(x)}$

At least on my system there is virtually no difference in the size of the sub/superscripts. The math markup, though, renders in a blurred style, as opposed to the html, which looks as text should look. The point with spacing is not a real argument either. IF need be, there are things like | | (|&thinsp;|).

In conclusion, I'm not convinced why one should (or has to) use math markup in the standalone formulas. Since this seems to be a topic which rests on some aesthetic feelings of all editors, it also seems that it is difficult or impossible to logically convince why one or the other variant is better. I think this is why the general peacemaking guideline "leave markup as it is, unless consensus is against" is in place, and this should be applied here. Jakob.scholbach (talk) 14:55, 18 November 2010 (UTC)

A few remarks:

I think that in the first place we should strive for overall article consistency and I don't think that allows for the coexistence of sections with HTML-formatted formulas (e.g. Logarithm#Logarithm of a product) and sections that cannot be formatted in HTML (e.g. Logarithm#More efficient series and Logarithm#Integral representation of the natural logarithm ).

I also think that the mixed formatting in the examples below is really unacceptable:

Example 1:

The following formula relates the logarithm of a fixed number x to one base in terms of the one to another base:

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

This is actually a consequence of the previous rule, as the following proof shows: taking the base-k-logarithm of the above-mentioned identity,

x = b^log_b(x),

yields

log_k(x) = log_k(b^log_b(x)).

Example 2:

Calculation of powers are reduced to multiplications and look-ups by

c^d = b^{(log_bc) · d}.

Divisions and roots are also covered by these two techniques since

${\frac {c}{d}}=c\cdot d^{-1}=b^{\log _{b}c-\log _{b}d}$

and

${\sqrt[{d}]{c}}=c^{\frac {1}{d}}$.

Furthermore, the math-equations are much easier to create, edit and maintain. Going back to your example, I think that maintaining

(''b''^{log_''b''(''x'')})^''p'' = ''b''^{''p'' · log_''b''(''x'')}

producing (b^log_b(x))^p = b^{p · log_b(x)},

is much harder and tiresome than

<math>(b^{\log_b(x)})^p = b^{p \cdot \log_b(x)}</math>

producing $(b^{\log _{b}(x)})^{p}=b^{p\cdot \log _{b}(x)}$.

DVdm (talk) 15:30, 18 November 2010 (UTC)

I don't get it why it is unacceptable to have a change of format in standalone formulas, but acceptable to have different markup for standalone vs. in-text math symbols. For your second point, there are arguments in favor of HTML and tex markup. We don't need to discuss them here. The only question is: is the current way of writing the article "unacceptable" (to quote you) by some guideline? If, on the contrary, it is up to the discretion of the editors who wrote the article (which is mostly me in this case), then there is no mandate to change it.

I sincerely hope that we can simply leave the discussion at this point. The likely result of a continued discussion will be at least one editor frustrated by this nonsensical wiki-editing question, + a huge waste of time. This time could be better invested, for example in reviewing the GA nomination of this article. I jumped over my shadow (as we say in German) for complex numbers. I heartily invite you to do the same here. Instead of picking out formatting issues, you would help this article and Wikipedia much more by pointing out, say, mathematical gaps, check the article for accessibility for lay readers, point out inconsistencies in presentation, ways to improve wording etc. That's what moves WP forward. Jakob.scholbach (talk) 19:23, 18 November 2010 (UTC)

(outdent) A couple of things on this, quoted as it's far from where you wrote it

"The guideline you do cite does not mandate your point. I think you misinterpret or overinterpret it. If everything in an article should have the same style, then you would have to replace html markup in any mathematical expression, not only standalone equations."

The guideline does mandate consistency, as a guiding principle no less. As for how to interpret it with regards to inline formulae it's discouraged, at MOS:MATH#Using LaTeX markup, with explicit guidance to handle inline formulae and standalone formulae differently. The consistency there is with the surrounding text, so e.g. √π is better than ${\sqrt {\pi }}$ as it disrupts the flow of the text less. On your most recent edit complying to the manual of style is one of the good article criteria, so can't be overlooked if you're looking to improve the article, while improving the formatting to make it more consistent and readable will help editors assess its other qualities more easily.

It's worth noting that this is totally under the control of the editor, as if they think an inline formula is too complex to display with HTML it can and perhaps should be made into a standalone formula, often using simpler and clearer markup when doing so. The reverse can be done for expressions that are simple enough and which don't need to be on their own. As this is largely determined by the surrounding text it's up to the editor currently working on it.--JohnBlackburne^words_deeds 19:55, 18 November 2010 (UTC)

After (ec). Jacob, to "point out inconsistencies in presentation" is exactly what we have in mind. I think that the presentation of this article is in a horrible state. If I were a lay reader, I would run away from it as far as possible, and if I would have some say in its fitness to be a GA, my say would be no, not as long as it looks like this. Yes, we do need to discuss markup arguments here. That is what an article talk page is for. If you like to discuss other aspects of the article, then by all means go ahead, as there is plenty of room here. But this is about format, and we already know your view, and we know mine, and John's, and the WP:MOS#general principles on WP:MOS#Internal consistency: "An overriding principle is that style and formatting choices should be consistent within a Wikipedia article, though not necessarily throughout Wikipedia as a whole. Consistency within an article promotes clarity and cohesion". I think that this is not open for interpretation or discussion, but it would be good to hear some other voices. Any takers? DVdm (talk) 20:21, 18 November 2010 (UTC)

With all due respect, you seem to be losing fair judgement when calling this article's markup "horrible" and that a lay reader would run away. This is clearly exaggerated—a lay reader would hardly ever care about such things, but much more so about whether the mathematical presentation is understandable. That's what I meant. Also, you might have noticed that I did post (see above) requests for substantive input on this article, but did not get anything. I think, your qualms about the GA-ibility of this article are also unfounded. The FA candidacy of groups, for example, which did include a lot of nitpicky technical formatting questions, never brought up such requests as your one, let alone people calling this "horrible".

I posted a request for comment at WT:WPM. I'm not around over the weekend, so may not be able to get back. Jakob.scholbach (talk) 21:05, 18 November 2010 (UTC)

Outside view

I came here after seeing Jakob's request at WT:WPM. Let me start with some things that I think everyone here agrees on:

1. The output of <math> tags renders differently from plain HTML.
2. In running text, the output of <math> tags may be too large compared to the surrounding text.
3. TeX notation is easier to edit and maintain.
4. <math> tags produce images, which load and render more slowly than HTML.
5. HTML is usually more accessible than <math> tags for users with disabilities.
6. Highly inconsistent formatting may be visually jarring.

Am I right in thinking that these are generally agreed upon? If not, stop me now.

Going on: If I understand the dispute correctly, it sounds like the following is true:

1. Broadly speaking, DVdm and JohnBlackburne prefer the output of <math> tags for displayed equations.
2. DVdm and JohnBlackburne believe that, when comparing nearby displayed equations, most changes between HTML and <math> tags are visually jarring.
3. Jakob.Scholbach believes that either HTML or <math> tags are acceptable for displayed equations.
4. Jakob.Scholbach believes that, when comparing nearby displayed equations, most changes between HTML and <math> tags are not visually jarring.

Yes? Again, if I'm not right here, please stop me now.

Assuming that I got all of that right, then what we have is really a disagreement over aesthetics. We can't solve aesthetic questions like this using purely rational, axiomatic criteria, so we have to approach the problem from a different angle.

I'm not going to propose a specific solution just yet, as I want to verify that I did in fact get all of the things I listed above correct. Did I? Ozob (talk) 00:23, 19 November 2010 (UTC)

I would quibble with your 5th point, that HTML is generally more accessible: I would say the opposite as HTML, at least when used for the math formulae it is capable of rendering, tends to produce text that is too small text to read with poor spacing, so is difficult for the visually impaired to read, while the workaround of coloured text is problematic for colour-blind readers. If the concern is over images being incompatible with screen readers the TeX renderer adds alt text of the raw TeX, probably more useful than the HTML which has little relation to the mathematical content.

On the second list although I do generally prefer TeX rendering that's not been my argument as I try and avoid using 'I like it' as a reason in any WP discussion. Instead as in this case if I think something is against generally accepted WP practice I try and find what the policy or policies on it are, and then follow them, even if sometimes that does not match my own instincts.--JohnBlackburne^words_deeds 01:22, 19 November 2010 (UTC)

Regarding accessibility, I had several things in mind. I know a couple of people who have severely impaired vision, and they have tools to zoom in on parts of their screen. For them this discussion is not relevant, because they need their zoom no matter what. However, people with moderately impaired vision may prefer to simply increase the font size. For them HTML is preferable because it scales smoothly and easily, without even creating an account. (Heck, I keep the font a little big even though my vision is fine; I think the default font around here is ridiculously small.) Increasing the browser's font size does not scale the images generated by <math> tags at all; I don't know how to scale the images. Finally, I don't have any experience with screen readers, but my expectation is that raw TeX is almost useless to the uninitiated and that HTML is more helpful; it would be helpful to get an opinion from someone with experience, however.

I agree that you are not using point one of my second list as your main argument (though it does seem to be in the background). I tried to capture your main argument in point two. If I understand you correctly, you have mainly spoken in favor of consistency. As Jakob.Scholbach has noted, however, this is a very slippery thing, because if I understand you correctly you are not advocating <math> tags in running text. But if we're not converting wholesale to <math> tags, then we have to decide: When are we consistent and when are we not? I may have misunderstood you, but I thought that you were arguing mostly for visual consistency; that's where my point two came from. If I've missed something then please let me know. Ozob (talk) 03:24, 19 November 2010 (UTC)

Thanks, Ozob, for helping out. I agree with all of your points. Maybe with a slight ε-ish restriction for your 3rd point (maintenance of tex vs. html). While the formulas in html do get longer, one does not need to know TeX in order to get things working. For articles which are less specialised such as this one, there may be a portion of potential editors who don't know TeX. But, again, this is only a side thought, and discussing this point would probably lead us astray from the general issue, I guess. Jakob.scholbach (talk) 08:05, 19 November 2010 (UTC)

P.S. TeX-ified equations do scale on my system when I change the browser's font size (Mozilla). But also, at least in this browser, there is (as mentioned above) no real difference in the size (even of iterated sub/superscripts). Jakob.scholbach (talk) 08:08, 19 November 2010 (UTC)

Ozob, thanks for the input. I think you summed it up pretty nicely, although for me point 1 of the second list —the one I guess you see as the personal list, so to speak— should definitely not be an argument in this. My main argument is related to point 2 of that list: full consistency within an article, and that is not a personal but a policy argument. In most formula-rich articles TeX is used for all standalone formulas, whereas HTML is only used inline for specifying some of the variables and for trivial stuff. Furthermore, and I am a bit reluctant to say this, but I also think that a 5th personal point should be added to the second list:

5. Jakob.Scholbach has put a lot of work in HTML-formatting formulas and finds our proposal to bring consistency into the article disrespectful (See comment above "I kindly, but firmly ask you to respect other people's work. I invested a lot in logarithms, which does not mean it is a perfect article, but which does mean that quite some thought has gone into this article, including markup".)

Apart from the formatting issue I do appreciate Jakob's work, but, firmly asking someone not to touch his work because a lot of work has been invested in it sounds a bit like a declaration of wp:ownership. (Jacob, please don't take this as an accusation, but rather as a friendly warning). DVdm (talk) 08:29, 19 November 2010 (UTC)

Maybe my formulation ("respect other people's work") was unfortunate. Of course, I did (and do) not mean that this article should never be touched or never be reformatted and I apologize if this didn't come across. More concisely: if a broad consensus should emerge from this discussion that HTML standalone formulas have to be replaced by TeX, I'm ready to go with it and will help out in making this change. However, I think this is unlikely to happen. Jakob.scholbach (talk) 08:46, 19 November 2010 (UTC)

Ok, no problem. By the way, Jacob, please note that I am not in any way asking you to make the changes. I am perfectly willing to fix the formatting myself along the lines of my first edit. Seems like a nice little weekend job, and I really don't mind doing that :-) Cheers - DVdm (talk) 09:06, 19 November 2010 (UTC)

Not much more to add except I see the point on accessibility, but I too have a browser (Safari) that scales images at the same time and by the same amount as text, as well as a shortcut (ctrl-mousewheel) that zooms the whole display. I suspect anyone for whom this is an issue will use a combination of software (OS, browser), settings and display hardware that adequately deals with the issue when needed, so it is probably not a big issue either way.--JohnBlackburne^words_deeds 22:38, 19 November 2010 (UTC)

Rendering equations as black text on white, or not as it happens

I just want to show an example that demonstrates the fact that actually Wikipedia math images are not rendered as black text on a white background. By taking the equations from this thread and placing them on a colored background you can see that some of the images are black text on white, but others are black text on a transparent layer. I don't know why Wikipedia mixes and matches like this although I guess that all images used to be black-on-white, but at some point the developers decided to use transparent images instead, but images that have already been rendered in the past are not re-rendered but cached versions are used.

$|z_{1}+z_{2}|=5$ but this $|z_{1}+z_{2}|=5\,$ which is indeed the same as this $\displaystyle |z_{1}+z_{2}|=5$ It's why after a few years of experimentation black text on a white background is used everywhere, and why we have a guideline, WP:COLOR. (blogb(x))p = bp · logb(x). vs. $(b^{\log _{b}(x)})^{p}=b^{p\cdot \log _{b}(x)}$ Example 1: The following formula relates the logarithm of a fixed number x to one base in terms of the one to another base: $\log _{b}{x}={\frac {\log _{k}{x}}{\log _{k}{b}}}.\,$ This is actually a consequence of the previous rule, as the following proof shows: taking the base-k-logarithm of the above-mentioned identity, x = blogb(x), yields logk(x) = logk(blogb(x)). Example 2: Calculation of powers are reduced to multiplications and look-ups by cd = b(logbc) · d. Divisions and roots are also covered by these two techniques since ${\frac {c}{d}}=c\cdot d^{-1}=b^{\log _{b}c-\log _{b}d}$ and ${\sqrt[{d}]{c}}=c^{\frac {1}{d}}$. Furthermore, the math-equations are much easier to create, edit and maintain. Going back to your example, I think that maintaining (''b''<sup>log<sub>''b''</sub>(''x'')</sup>)<sup>''p''</sup> = ''b''<sup>''p'' · log<sub>''b''</sub>(''x'')</sup> producing (blogb(x))p = bp · logb(x), is much harder and tiresome than <math>(b^{\log_b(x)})^p = b^{p \cdot \log_b(x)}</math> producing $(b^{\log _{b}(x)})^{p}=b^{p\cdot \log _{b}(x)}$.

To my eyes it looks like all images are meant to be on white backgrounds as even those without a white rectangle have antialiasing to a white background around their edges. You can use coloured text (see Help:Formula#Color) so presumably not background, but as all WP styles are on a white background this is not much of a problem that I can see. It stops editors going overboard with coloured backgrounds in tables and the like, but this is probably a good thing.--JohnBlackburne^words_deeds 23:27, 20 November 2010 (UTC)