Talk:Logarithm

Article Checklist for "Logarithm"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developed article: complete or nearly so
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Fredrik Johansson 18:02, 28 April 2007 (CDT)

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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)

definition

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. $log_{2}(x)$. --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)