Talk:Logarithm
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Sources for "Feynman's algorithm"
[edit]This entire sections seems to be based on a single, vague, two paragraph description in Physics Today (here), with the details being fleshed out through original research. Physics Today may a reliable source, but the article in question is obviously meant for a popular audience, and so is extremely lacking in detail, and does not cite any sources itself from which further detail might be found. If someone can find an academic paper, textbook or reference book that describes this algorithm then I'd be convinced that it belongs in Wikipedia, but as it stands I find its inclusion highly dubious at best. The section title might be taken to imply that the method is known as "Feynman's algorithm" in the literature, but can find no evidence that anyone aside from the editor who added this section called it that. If have no doubt that the method is, in fact valid, but this does not alone merit its inclusion since much of the detail seems to be the result of original research. --RDBury (talk) 03:11, 15 October 2020 (UTC)
- It seems that this is the first attempt (or one of the first attempts) for computing logarithms on a computer, and that it is its only interest, as it seems to be not specially efficient. So I suggest to reduce this paragraph to a single line, and to move it to the history section. D.Lazard (talk) 03:53, 15 October 2020 (UTC)
- @D.Lazard: The algorithm, according to the PT article, was from when Feynman was still at Los Alamos, at which time "computer" meant a person working the computations by hand. The fact that the algorithm itself seems to be optimized for a binary (electronic) computer would seem to contradict this though. The algorithm does seem seem efficient since it eliminates multiplication and division from the computation, but whether more efficient methods are used by modern processors I can't say. There's a trade-off between look-up table size and number of computations, so it's hard to day what 'efficiency' actually means here. For the history section I think we have a similar issue: is there a reference which shows the historical significance of the algorithm? --RDBury (talk) 04:16, 19 October 2020 (UTC)
- At Los Alamos, Feynman had some IBM machines that weren't quite computers. As well as I know, the fanciest of them could read two numbers off a punched card, multiply them, and then punch the product onto the card. (Especially useful for payroll calculations.) So finding algorithms to do complicated calculations using primitive equipment, might have come up with some interesting algorithms. Gah4 (talk) 16:37, 4 November 2021 (UTC)
- @D.Lazard: The algorithm, according to the PT article, was from when Feynman was still at Los Alamos, at which time "computer" meant a person working the computations by hand. The fact that the algorithm itself seems to be optimized for a binary (electronic) computer would seem to contradict this though. The algorithm does seem seem efficient since it eliminates multiplication and division from the computation, but whether more efficient methods are used by modern processors I can't say. There's a trade-off between look-up table size and number of computations, so it's hard to day what 'efficiency' actually means here. For the history section I think we have a similar issue: is there a reference which shows the historical significance of the algorithm? --RDBury (talk) 04:16, 19 October 2020 (UTC)
It seems that Feynman rediscovered the "radix method", e.g. as described in "RADIX METHOD OF CALCULATING NATURAL LOGARITHMS IN BINARY NOTATION", which I can't actually find a copy of. But similar things are done with decimal radix as here. I think Danny Hillis's referenced brief description of the method is adequate, and the adaptation of it in the article is even more clear and correct. Dicklyon (talk) 03:11, 17 November 2021 (UTC)
Cent scale
[edit]Okay, I'm a "noob" at editing Wikipedia pages (this is my second attempt in 15 years). So here goes:
In the "Particular bases" section of the Logarithm page, it says:
". . . in music theory, where a pitch ratio of two (the octave) is ubiquitous and the cent is the binary logarithm (scaled by 1200) of the ratio between two adjacent equally-tempered pitches in European classical music; . . ."
If you look at the Wikipedia definition of "Cent (music)" [1], and the Wikipedia definition of "Binary logarithm", it appears that there has been a transcription error between these two definitions [Cent (music) & Binary logarithm] and the Logarithm article, which may need to be harmonized.
Based on my music theory classes in college, a cent is 1/100th of the pitch ratio between two adjacent equal-tempered pitches (a interval of a semitone or half-tone) in European music, and 1200 cents make an octave - not a half-tone.
So should the above quoted text in the Logarithm article read: ". . . and the cent is the binary logarithm (scaled by 100) of the ratio between two adjacent equally-tempered-pitches in European classical music; . . ."?
Or would another way of re-writing this be better?
Or am I completely mistaken?
TimeriderTech (talk) 01:10, 4 November 2021 (UTC)
- I think it's more screwed up than you indicate. It should perhaps say ". . . in music theory, where a pitch ratio of two (the octave) is ubiquitous and the cent is the binary logarithm (scaled by 1200) of the ratio between two pitches; . . .". Any two pitches, not two notes nominally 100 cents apart. On the other hand, this is still a bad way to define "the cent" where it means "the interval measured in cents". Please try to rewrite it to make more cents. Dicklyon (talk) 01:17, 4 November 2021 (UTC)
- It's just overly specific. The number of cents between any two pitches is 1200x the binary logarithm of their ratio. There is no reason to specify that the two pitches in question be adjacent. —David Eppstein (talk) 01:21, 4 November 2021 (UTC)
- What you say is correct. But it's hard to incorporate it into a sentence that says "the cent is". Dicklyon (talk) 03:24, 4 November 2021 (UTC)
- What's wrong with my proposed simpler replacement for the incorrect quoted text above: "the number of cents between any two pitches is the binary logarithm (scaled by 1200) of their ratio" ? —David Eppstein (talk) 07:13, 4 November 2021 (UTC)
- Nothing wrong with that. I interpreted your "It's just overly specific" as a suggestion to make the smaller change like I first said, with "the cent is". An alternative using would be "The cent is 1/100 of the difference of logarithms between two pitches a semitone apart" or something like that, where now the log base doesn't even matter. It's exactly the same problem discussed at length on Talk:Decibel, between trying to say "the decibel is" or trying to say how many decibels a ratio corresponds to. The standards agencies take the former approach, but I'm not saying I like it. Dicklyon (talk) 14:36, 4 November 2021 (UTC)
- What's wrong with my proposed simpler replacement for the incorrect quoted text above: "the number of cents between any two pitches is the binary logarithm (scaled by 1200) of their ratio" ? —David Eppstein (talk) 07:13, 4 November 2021 (UTC)
- What you say is correct. But it's hard to incorporate it into a sentence that says "the cent is". Dicklyon (talk) 03:24, 4 November 2021 (UTC)
- It's just overly specific. The number of cents between any two pitches is 1200x the binary logarithm of their ratio. There is no reason to specify that the two pitches in question be adjacent. —David Eppstein (talk) 01:21, 4 November 2021 (UTC)
Thank you both for chiming in on this! My attempt to create a new "Cent scale" sub-topic on the talk page for "Logarithm" didn't appear to go as planned, and I'm guessing that somebody silently helped me out. (Thanks! I promise I'll get better at editing.) It also makes me feel better that at least two other editors agree that there's a problem here. Thanks to you both! I'm going to ruminate on a better re-write of the erroneous sentence, incorporating the above input. Hopefully we can all find an alternative phrasing that corrects this mathematical error in a manner most pleasing and clear to read. 64.201.116.75 (talk) 16:33, 4 November 2021 (UTC)
References
Units and logarithms
[edit]Really need a section on the units of a logarithm. For example, what is the unit of Log(10/seconds)? That is a common expression in first order rate equations, like nuclear decay.
From the integral definition of the natural log, "the area under the curve of a plot of 1/x versus x", it is clear that x can have any unit and that the logarithm is always unitless. The "area" under the curve has units of the x-axis times the units of the y-axis which, in all cases, is unity and unitless.
There are some discussion on the web suggestion things like "you can't take the logarithm of a number with units" (which is absurd, scientists and engineers do it all the time), to basically "there is a hidden and highly secret process in which the units disappear". Like "actually log(x*unit) is really log(x*unit/1*unit) so the units cancel", which is wrong.
The fact that the logarithm removes the units means that taking a logarithm is a lossy transform. There is no way, other than external knowledge, that allows the unit to be recovered by taking the exponent of the log. Therefore, e^(Ln(x)) <> x in all cases since the units have been lost. Jsluka (talk) 18:49, 26 October 2022 (UTC)
- Engineer here - Just because we often do things like take the logarithm of numbers with units does not mean we should. For example, expanding something like log(x*unit) into log(x) + log(unit) implies a solution x to base ^ x = unit (definition of a logarithm), which is really hard to find when the base is in R (like, say, Euler's constant) and you're confined to R or C for the solution x. I don't know how to raise a real number to another real (or even complex) exponent in a way that changes the units from 1 to something like seconds, square meters, or volts. I think we can take logarithms of units when we define them -- Otherwise, I would personally consider something like ln(x) -> R; where x = seconds (in somewhere like R or C) completely nonsense unless one or more of three things is the case:
- There is a sensible solution to the expression exp(x) = seconds, for x in your given space. Good luck.
- The argument in question winds up being dimensionless by cancellation.
- We can "pretend" the units are dimensionless because we plan to just call the units of the solution something like log-seconds anyways.
- Taking this in a weird direction, it means that we can make equivalence relations between log units in some weird ways. Consider ln(acceleration) == ln(m/s^2) == ln(meters) - 2*ln(seconds), which suggests that ln(seconds) == (ln(meters) - ln(acceleration))/2 == ln( (m / (m/s^2)) ^ (1/2) ) == ln( (seconds^2) ^ (1/2)), which does make sense. I don't know what to do with a result like this, but it does serve your point that clearly one can take the logarithm of a number with units. In fact, one can use a logarithm of pure units not multiplied by anything at all and arrive at sensible answers, just like this. 66.218.139.214 (talk) 19:37, 24 April 2023 (UTC)
- Reminds me that it seems to me that engineers often write expressions and equations with units factored out, while physicists keep units in. That is, one might say F(Newtons) = m (kg) * a (m/s/s). The expectation is that one converts to the given units, applies the formula, and then converts the result. Physics will just say F = m a, where the variables keep the units. The former might result in excess conversions, the latter unusual units. Since computer languages normally don't keep units in variables, the engineering way is convenient for programming. And, as with the question here, units go out before the logs. Gah4 (talk) 04:55, 25 April 2023 (UTC)
Redundant formula in section on Taylor series
[edit]Jacobolus surprisingly reverted an edit that removed a verbatim copy of a line two lines above that line. I'm re-reverting it after [Discussion] with him. MüllerMarcus (talk) 20:21, 20 October 2023 (UTC)
- No, please do not make excellently accessible featured articles less friendly to novice readers without consensus, and please don't engage in edit warring. The explanation is not "redundant", and the line you call "*identical*" is self evidently different. Please look again more carefully, and try to put yourself in the shoes of a non-technical reader. –jacobolus (t) 20:39, 20 October 2023 (UTC)
- I made the text a bit more explicit. Does that help? (As always, anyone should feel free to keep working on this who thinks they can improve on it.) –jacobolus (t) 00:58, 21 October 2023 (UTC)
- that really helps a lot! Thank you for spending time on this, and sorry for the confrontation. MüllerMarcus (talk) 12:21, 21 October 2023 (UTC)
Exponentiation's other inverse function?
[edit]Isn't root extraction also an inverse function of exponentiation? - S L A Y T H E - (talk) 09:04, 25 July 2024 (UTC)
- No, it is an inverse function of a power function with an integer exponent D.Lazard (talk) 10:24, 25 July 2024 (UTC)
- Well, first, power function allows for non-integer exponents. But ok, power function redirects to exponentiation, and exponentiation includes both power function and exponential function. The important difference, and I don't know which one you are asking about, is between exponential function and exponentiation. Gah4 (talk) 20:01, 25 July 2024 (UTC)
- To be clear, inverse functions are defined for univariate functions only, and exponentiation is a bivariate function. So, for having an inverse function, one must fix one variable (partial application). If one fixes the base, one has an exponential function with a logarithm as an inverse function. If one fixes the exponent, one has a power function. If the exponent is a natural number n, the inverse function is a nth root. D.Lazard (talk) 21:17, 25 July 2024 (UTC)
- Following Lazard (25 July 2024) we can conclude that the first sentence of the lede is wrong - since 2015. The definition shouldn't speak of the inverse function of the bivariate operation 'exponentiation'
- I suppose that was meant: a logarithm is a (written) expression with a symbol for a certain logarithmic function, followed by a symbolic expression for a domain element. Combined with:
- A function f on R+ is called logarithmic when it transforms multiplication into addition: f(uv) = f(u) + f(v) . Or - in an indirect way - when it is the inverse function of an exponential function of type x→bx (b>0, ≠1) .
- Or anything equivalent. Yes? Hesselp (talk) 19:47, 21 August 2024 (UTC)
- This seems excessively pedantic and confusing, but it's plausible we could make up a better first few paragraphs.
- If you want to be precise, the logarithm function is the inverse of the exponential function (or a logarithm function is the inverse of an exponential function). The term logarithm, most precisely, is a synonym for exponent, but saying it that way can also be confusing to novices. The unadorned term logarithm is also routinely applied to the logarithm function, or to an expression such as or . –jacobolus (t) 20:13, 21 August 2024 (UTC)
- A logarithmic function (or logarithm function) f (R+→R) transforms multiplications into additions, that is: f(uv) = f(u) + f(v) for all pairs u, v of positive numbers.
- Is this simple and clear enough to start the lead with? Hesselp (talk) 19:05, 24 August 2024 (UTC)
- That is a useful property of the logarithm function, but it does not describe what the function is. I don't see what's wrong with the current lead. Each exponential function has an inverse logarithm function . The log is the inverse of the exponential. The fact that there is a family of exponential functions, and a corresponding family of logarithm functions, is detailed later.
- You could also define a different two-variable powering function rather than considering a family of single-variable exponential functions but that does not have a well-defined inverse as it is not even locally 1-1 from its inputs (the pair ) to its outputs (a single number). —David Eppstein (talk) 19:27, 24 August 2024 (UTC)
- An alternative would be to say something like: "In the equation the quantity is called the base, the quantity is called the exponent, and the quantity is called the result. When this equation is rewritten to isolate the exponent , is instead called the logarithm base of , denoted ." But this is probably just as (if not more) confusing. [Edit: the § Definition section already discusses this.] –jacobolus (t) 19:44, 24 August 2024 (UTC)
- A logarithmic function (or logarithm function) f (R+→R) transforms multiplications into additions, that is: f(uv) = f(u) + f(v) for all pairs u, v of positive numbers.
- Following Lazard (25 July 2024) we can conclude that the first sentence of the lede is wrong - since 2015. The definition shouldn't speak of the inverse function of the bivariate operation 'exponentiation'
- To be clear, inverse functions are defined for univariate functions only, and exponentiation is a bivariate function. So, for having an inverse function, one must fix one variable (partial application). If one fixes the base, one has an exponential function with a logarithm as an inverse function. If one fixes the exponent, one has a power function. If the exponent is a natural number n, the inverse function is a nth root. D.Lazard (talk) 21:17, 25 July 2024 (UTC)
- Well, first, power function allows for non-integer exponents. But ok, power function redirects to exponentiation, and exponentiation includes both power function and exponential function. The important difference, and I don't know which one you are asking about, is between exponential function and exponentiation. Gah4 (talk) 20:01, 25 July 2024 (UTC)
"it does not describe what the function is" ??
[edit]@David Eppstein: Please can you motivate why (an - improved? - variant of Aug 24):
"A function f (R+→R) is called logarithmic function (or logarithm function) iff f(uv) = f(u) + f(v) for all pairs u, v of positive numbers."
you don't see as a precise definition?
Is the definition only precise, using the inverse function?:
"A function f (R+→R) is called logarithmic function (or logarithm function) iff finv(u) + finv(v) = finv(uv). finv(u+v) = finv(u) · finv(v) for all pairs u, v of pos. numbers."
Or should it be?:
"A function f (R+→R) is called logarithmic function (or logarithm function) iff b ^ f(x) = x, b>0 ≠1, x>0 ."
presuming knowledge of the not very elementary (IMHO) bivariate operation ^ for real variables, instead of addition and multiplication.
@Jacobolus: Why "the quantity b, x, y" ? This letters are used here as variables. They all three represent real numbers, not more general quantities.
The definition of "logarithm base b of y" really needs a mysterious rewriting of an equation? There really isn't a more direct way?
And after all I don't see a definition of logarithmic function / logarithm function. Hesselp (talk) 12:55, 25 August 2024 (UTC) Hesselp (talk) 14:40, 25 August 2024 (UTC)
- I think all of your proposals are less clear or helpful for the lead section of this article than the existing text. Conflating "logarithm" and "logarithm function" is common and not really a serious problem. The distinction could be mentioned somewhere in a footnote or later down the article but I think belaboring it at the start is distracting. Anything involving symbols such as does not belong in the lead section of this article. –jacobolus (t) 15:07, 25 August 2024 (UTC)
- I have edited the beginning of the lead for avoiding the ambiguous (here) "exponentiation", using a formulation proposed above by Jacobolus (I read Jacobolus' formulation only after having edited the lead). D.Lazard (talk) 15:16, 25 August 2024 (UTC)
- I think a link to exponentiation is more helpful to provide context here for most readers than exponential function. –jacobolus (t) 15:27, 25 August 2024 (UTC)
- I have restored this link, using the fact that in Exponentiation, the base is defined in the second sentence. D.Lazard (talk) 15:54, 25 August 2024 (UTC)
- I think a link to exponentiation is more helpful to provide context here for most readers than exponential function. –jacobolus (t) 15:27, 25 August 2024 (UTC)
Definition of logarithm
[edit]This is vaguely on the topic of the last discussion. I note that the article doesn't really properly define logarithms. The section on the definition is a standard one used in pre-calculus, but it is not suitable for analysis, which uses properties of the natural logarithm instead. Thus, really is defined as , where is defined as an integral (or in some other equivalent way). The problem with the definition "The real such that " is that the exponential is usually defined as , so it is circular, unless we except the natural logarithm and exponential function. Tito Omburo (talk) 16:32, 25 August 2024 (UTC)
- Explaining this is fine, but it should be done below the first half of this article. –jacobolus (t) 17:30, 25 August 2024 (UTC)
- Yes, I agree. Particularly since the definition given in the article already isn't "wrong" per se, it's just logically incomplete. Tito Omburo (talk) 20:17, 25 August 2024 (UTC)
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