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 Definition Inverse of exponentiation, as subtraction is the inverse of addition and division is the inverse of multiplication. [d] [e]
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Logarithm article

I just wrote this today. It may be a little too elementary. I hope others will add to it and edit it. --Catherine Woodgold 17:09, 28 April 2007 (CDT)

...and check whether my math is correct (thanks, Fredrick Johansson -- and thanks again for the graph!) --Catherine Woodgold 18:44, 28 April 2007 (CDT)

Notational variants

I added a section on the different notations that are commonly used to indicate the base. It looks a little out of place where I put it, so plese move it if you think there's a better location in the article. Greg Woodhouse 22:47, 28 April 2007 (CDT)

I see what you mean. It's not easy to find the best place for it. It could go almost anywhere, including right after the introduction, or at the end. Not between "Extension of logarithms to fractional and negative values" and "Shape of the logarithm function", though, as that would disturb the connection between those two sections. (Perhaps a sign of a well-written article would be that no two consecutive sections could have anything added between them without disturbing the flow.) Another possibility would be to combine the notational information into the introduction; yet another would be to move some of the information out of the introduction into the notational section. I've moved the notational section to be the second-last section just before the complex number section, but if someone wants to move it again, feel free. --Catherine Woodgold 08:17, 29 April 2007 (CDT)

Extension of logarithms to non-natural number exponents

Re this part: "Rules for adding and multiplying exponents were noticed, and to extend the idea to fractions and negative numbers it was assumed that the same rules would apply." This is an intuitive, non-rigourous explanation. I haven't shown that it's possible to do that in a consistent, meaningful way. I'm not sure whether I can figure out how to do it rigourously. Perhaps someone who knows how could help here. I'd like to keep the intuitively appealing explanation and either fix it up to make it rigourous, or add a rigourous explanation separately, or state that it's not rigourous and tell the reader where to find a rigourous definition/construction of logarithms in a bibliography (which I'm hoping others will eventually supply). --Catherine Woodgold 07:55, 29 April 2007 (CDT)

Stuff to add, possibly

  • Use of logarithms to do multiplication using tables (or slide rules).
  • Use of logarithmic scales on graphs
  • Taylor series of the natural logarithmic function (or just a link to the Taylor series page, if it give it).

--Catherine Woodgold 20:33, 5 May 2007 (CDT)

The points 2 and 3 seem to be good ideas! I think we could do without description of multiplication using tables as it is somewhat exotic nowadays (I was taught about it but it was long ago; now, I wouldn't teach it). It would be interesting, however, just to mention that application in its historical context. Perhaps more technical details could be put in slide rule article. --Aleksander Stos 15:49, 12 May 2007 (CDT)


The present intro (A logarithm is a mathematical function which provides the number which would appear as the exponent in an expression) seems to be clear only for those who already know what the logarithm is. In other words, "providing the exponent for _an_ expression" is correct but too general. I'd suggest something like this: "A logarithm is an elementary mathematical function which is inverse to the exponential function of a given base, i.e. it returns the number which would appear as the exponent in the latter". Native speakers surely would find a better formulation. --Aleksander Stos 15:31, 12 May 2007 (CDT)

I'm not sure the first sentence should be an attempt to give a mathematical definition. Something along the lines of "converts multiplication into addition" with an allusion the historical significance of logarithms might be more helpful. Fredrik Johansson 15:45, 12 May 2007 (CDT)
I agree that the first sentence does not need to be a mathematical definition. BTW, my proposition wasn't either :). My only concern is that providing the exponent in an expression means not that much. I like to add some specific context, that's all (the formulation can be modified as you like). --Aleksander Stos 16:03, 12 May 2007 (CDT)
My instinct is that the definition of a logarithm as the inverse function to an exponential ("undoing" the exponential) is both the most fundamental and the most easily understood way to introduce them. The current first sentences would work well right after a new first sentence in this vein. The fact that it "converts multiplication to addition" is not a fundamental part of the definition of a logarithm, but rather a happy side benefit; we could introduce it as such - and/or bring it up in relation to slide rules. - Greg Martin 16:05, 12 May 2007 (CDT)
How about "The logarithmic function or logarithm is the inverse of the operation of exponentiation."? --Catherine Woodgold 19:44, 12 May 2007 (CDT)
I think that works. Greg Woodhouse 19:53, 12 May 2007 (CDT)
I think "inverse of exponential" is clearer than "provides the number which would be in an exponent". I like the idea of using a non-mathematical statement for the introduction, but I'm not sure that is the right one yet. - Jared Grubb 21:36, 12 May 2007 (CDT)
When I was in grade school, subtraction was introduced as the inverse of addition, and so forth. It's not really a concept that should be that unfamiliar. Besides, anyone looking up logarithm should have at least some exposure to basic mathematics, as would anyone that knows what exponents are. Greg Woodhouse 22:21, 12 May 2007 (CDT)
Let me amend that -- I would say "A" logarithmic function rather than "The" logarithmic function. Each base gives a different function, e.g. . --Catherine Woodgold 07:41, 13 May 2007 (CDT)
I think it's not necessary to be that precise about the difference between logarithmic functions and logarithms. So, I'd simply say "Computing logarithms is the inverse of exponentiation". We can embellish it a bit, e.g., "Computing logarithms is the inverse of exponentiation, like subtraction is the inverse of addition and division is the inverse of multiplication"; I think that's quite accessible.
The alternative of not mentioning "inverse" leads to rather convoluted sentences. The best I can come up with now is "The logarithm of a number to a certain base is the power to which that base must be raised to obtain the given number." I think it's not too bad, and better than the current first sentence (if I may say so), but it's not easy to understand without re-reading the sentence. It's more precise though. By the way, I'm not sure that the first "to" in my sentence is the correct preposition. -- Jitse Niesen 08:23, 13 May 2007 (CDT)


I'm having trouble following the overall story in the article. For instance, the second paragraph says that log10(46) is 1.63347…, but fractional exponents are only introduced later. Similarly, I'd expect to see the rules and next to each other, but they are not. So, I think the first step is some overall planning.

The article spends quite some time talking about exponentiation: how to raise a number to a fractional or negative power. Is this the proper place? In my experience, this is usually taught before logarithms, but I can see the argument that it is so closely linked that it should be treated in detail here.

The third paragraph says it's more natural to use base e. But this is nowhere explained. Part of the reason is probably that the derivative of the natural logarithm is 1 / x; another fact noticed there which seems a bit out of place.

Regarding the stuff to add: the use of logarithmic scale in graphs could be an excellent example to make the subject alive for the nonspecialist. A bit on the history (tables, slide rules) would be nice, especially since logarithms have a rich history, but not necessary.

The Taylor series has lowest priority for me: it's a nice fact, but unless it can be blended in with the rest of the article it's no more than a fact. On the other hand, a lot of people looking up logarithms might be looking for a specific formula, like its Taylor series, or its derivative, or rules like . There's a tension here between writing a nice article and giving the readers what (we that that) they want.

The rule should probably be added. But I think the article has (almost) sufficient material for approval; it's mainly a question of rehashing it.

Sorry, not much specific guidance on how to improve the article, but perhaps some food for thought. -- Jitse Niesen 07:17, 18 May 2007 (CDT)

Thanks for your astute comments. If I or others can figure out a way to rearrange it according to those ideas it will improve the article considerably. I'll think about it, and anyone else is also welcome to try their hand. Re whether to include the material on fractional and negative exponents: perhaps one way to decide this would be to think about whether we're going to have a separate article on "exponents" or "exponentiation" and what would be natural to include there. There's already an article on exponential growth, containing material from Wikipedia. One possibility is not to have an article on exponents per se, just the page on exponential growth and to leave the material on negative and fractional exponents on this page. (Where would "exponent" redirect to in that case?) Another idea would be to have an article on "exponents" with an elementary explanation of what they are and the material from here about fractional and negative exponents. Maybe that would be too short an article; or maybe it would be fine, along the lines of the Least common multiple and Greatest common divisor pages. Another idea would be to include in the same article material about the exponential function and the number e. Actually, I'm starting to feel that that would be a good idea: beginning with an elementary explanation of exponents and working up to the discovery of the number e as a climax. I'm now leaning towards moving the material on fractional and negative exponents off this page. I'm interested to know what others think on this question. --Catherine Woodgold 06:00, 19 May 2007 (CDT)


In my opinion, haing n article on th exponential function is even more important (and certainly more fundamental) than an article on logarithms. Greg Woodhouse 13:10, 20 May 2007 (CDT)

Two very minor revisions

The first sentence in the article defines the logarithm as the inverse of "exponentiation" and assumes that all readers will understand the meaning of that word. I added a sentence to the lead-in paragraph to help explain what that word means.

I revised the last sentence of the section entitled "Logarithms of intermediate and fractional values" to clarify that the logarithms of numbers greater than 1 are positive real numbers as shown in the image of the logarithm function. The previous wording in that sentence about numbers greater than one was difficult to understand.

The two revisions were so minor that I saw no reason to first discuss them here. I hope that this does not offend anyone. Milton Beychok 18:08, 25 October 2008 (UTC)

Additional content about notational variances

I felt that it was important to add additional content to the section on notational variants to stress the importance of avoiding confusion by using notations which cannot be misunderstood. I did not change any of the content that was already in that section ... I merely added additional information. I also modified the section heading from "Notational variants" to "Bases and notational variances" since the different notations are due to different logarithmic bases. Milton Beychok 23:12, 27 October 2008 (UTC)

Reduced size of image to provide more width for text

Dmitri, I reduced the size of the graph somewhat so as to provide more width for the written text. I suspect that is originally why it was centered below the text before you moved it to the right of the text. The reduced size is still quite legible. Regards, Milton Beychok 23:18, 30 October 2008 (UTC)

You did fine, but now the caption looks too long. I go to shorten it, moving the sentence about the cut to the text. Dmitrii Kouznetsov 23:47, 30 October 2008 (UTC)
I replaced the reduced version of the original graph with a new smaller version yet (284 px instead of 375) so as to provide even more room for the article text. To do so, I had to redraw the graph completely and use larger text font inside the graph so that text would still be legible in the smaller version. The original graph is [[Image:Logarithms.png]] and the new smaller graph is [[Image:Logarithms2.png]]. Milton Beychok 23:19, 31 October 2008 (UTC)

Removed Category designations from bottom of the article's Edit page

Dmitri, as discussed on your user Talk page by Jitse Niesen and Chris Day, we do not create categories (as in Wikipedia) in Citizendium. Workgroup categories are listed in the Metadata template that is created when a new article is created. For example, the Metadata page for this article lists the Mathematics workgroup and the Engineering workgroup as as "cat"s.

Also, this article does have a definition and it is given in the blue and gray box at the top of this page. Notice that it has a [d] and an [e] at the end of the definition. Clicking on the [d] takes you to the Definitions subpage. Clicking on the [e] takes you to the Definitions edit page.

That blue and gray box also lists the two workgroups designated in the Metadata template.

I hope this helps you to understand the Citizendium system. It is quite different from the Wikipedia system. Regards, Milton Beychok 15:20, 31 October 2008 (UTC)

Adding a new section: "The Importance of logarithms"

I posted a message on the Talk page of Jitse Niesen (a mathematics editor) asking him if he thought it would be useful to have this article include some numerical examples of logarithmic calculations. Jitse said that he thought it would be useful. In view of the fact that calculators and computers have virtually eliminated using logarithms for arithmetic computations, he also suggested that discussion of the present day importance of logarithms would also be of interest.

Accordingly, I have created a new section on "The Importance of logarithms" which combines numerical examples of arithmetic computations (that were important and useful prior to calculators and computers) as well discussing the important current day applications of logarithms. Milton Beychok 04:28, 3 November 2008 (UTC)

Discrete logarithm

We have an article with that title. This one should probably link to it, but I'm not certain where the link would go, so I'm mentioning it here rather than just adding the link. Also, that article could likely use some attention from a better mathematician than its author (me). Sandy Harris 02:50, 4 January 2009 (UTC)

Sandy, I am adding a link to Discrete logarithm into the Related Articles subpage of this article. I think that is the appropriate page for related articles that are not directly mentioned in an article. When a Related Articles subpage for Discrete Logarithm is created by you or anyone else, it should probably include a link back to this Logarithm article. Regards, Milton Beychok 03:41, 4 January 2009 (UTC)