Talk:Exponential function

List of potential sources

Feel free to add to this list of some sources about the exponential function. Some of these are already cited by the article, and others may or may not be useful to cite. I'll also try to keep adding more to this list in the next few days: –jacobolus (t) 04:18, 14 November 2024 (UTC)

• Ahlfors, Lars (1966), "The Exponential and Trigonometric Functions", Complex Analysis (2nd ed.), New York: McGraw-Hill, §2.3, pp. 43–48, LCCN 65-20106
• Apostol, Tom M.; Mnatsakanian, Mamikon (1998), "Surprising geometric properties of exponential functions", Math Horizons, 6 (1): 27–29, doi:10.1080/10724117.1998.11975073
• Barnett, Janet Heine (2004), "Enter, Stage Center: The Early Drama of the Hyperbolic Functions", Mathematics Magazine, 77 (1): 15–30, JSTOR 3219227
• Bell, A. H. (1932), The Exponential and Hyperbolic Functions and their Applications, London: Pitman & Sons
• Cajori, Florian (1913) "History of the exponential and logarithm concepts", The American Mathematical Monthly 20 (1): 5–14; (2): 35–47; (3): 75–84; (4): 107–117; (5): 148–151; (6): 173–182; (7): 205–210, JSTOR 2973509; 2974078; 2973441; 2972960; 2972412; 2973069; 2974104
• Courant, Richard; Robbins, Herbert (1941), "The Exponential Function and the Logarithm", What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, §8.6, pp. 442–452, ISBN 0-19-502517-2
• Feeman, Timothy G. (2001), "Conformality, the exponential function, and world map projections", The College Mathematics Journal, 32 (5): 334–342
• Gasparini, Andrea; Key, Eric; Radcliffe, David (2019), "A Geometric Approach to the Natural Exponential Function", The College Mathematics Journal, 50 (5): 357–363, doi:10.1080/07468342.2019.1664846
• Gowers, Timothy (2008), "The Exponential and Logarithmic Functions", in Gowers, Timothy (ed.), The Princeton Companion to Mathematics, Princeton University Press, §3.25, pp. 199–202, ISBN 978-0-691-11880-2
• Huntington, Edward V. (1916), "An elementary theory of the exponential and logarithmic functions", The American Mathematical Monthly, 23 (7): 241–246, JSTOR 2973888
• Klein, Felix (1932), "Logarithmic and Exponential Functions", Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis, translated by E. R. Hendrick; C. A. Noble, London: MacMillan, §3.1 pp. 144–162
• Komornik, Vilmos; Schäfke, Reinhard (2024), "A Simple Introduction to the Exponential Function", The College Mathematics Journal, 55 (2): 165–168, doi:10.1080/07468342.2023.2234256
• Maor, Eli (1994), "⁠$e^{x}$⁠: The Function That Equals Its Own Derivative", e: The Story of a Number, Princeton University Press, Ch. 10, pp. 98–113, ISBN 0-691-03390-0
• McCartin, Brian J. (2006), "e: The master of all", The Mathematical Intelligencer, 28: 10–21, doi:10.1007/BF02987150
• Melzak, Zdzislaw A. (1975), "On the exponential function", The American Mathematical Monthly, 82 (8): 842–844, JSTOR 2319809
• Toth, Gabor (2021), "Exponential and Logarithmic Functions", Elements of Mathematics: A Problem-Centered Approach to History and Foundations, Springer, pp. 423–468, doi:10.1007/978-3-030-75051-0_10, ISBN 978-3-030-75050-3
• Yzeren, Jan van (1970), "A rehabilitation of ⁠$(1+z/n)^{n}$⁠", The American Mathematical Monthly, 77 (9): 995–998, JSTOR 2318122

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One more title for the list.
H.A. Lorentz (Nobel Prize 1902), Lehrbuch der Differential- und Integralrechnung, 3. Aufl. 1915, S. 44 :

Funktionen, bei denen die unabhänglige Variabele im Exponenten einer Potenz auftritt, wie zum Beispiel  $\,a^{x}$, $\,a^{px^{2}}$, $\,a^{\sqrt {x}}$,  nennt man exponentiellen Funktionen.

I'm quite sure that that more recent sources can be found as well, saying that functions with the independent variable occurring in the exponent of an exponentiation, are called 'exponential'.  With as (unintential) consequences: (1)  every (positive) function  f  is exponential, for it can be written as $\,x\mapsto e^{\ln f(x)}\,$.   And (2)  the function $\,x\mapsto \Sigma _{i=0}^{\infty }x^{n}/n!$  is not exponential.
Conclusion: a definition of 'exponential function' shouldn't be based on how it can be expressed in written form, but on an intrinsic, characteristic, property. Hesselp (talk) 17:11, 2 December 2024 (UTC)

I don't understand why you are dragging the conversation about possible definitions into this substantially unrelated section about gathering sources. I don't speak/read German. Other than calling a broad class of functions "exponential", is there some other insight in this book that isn't found in English language sources? (Here's a link to the relevant page of the 1900 edition). –jacobolus (t) 17:55, 2 December 2024 (UTC)

A gentler introductory section before diving into properties of exp

I've been working on draft new sections for this article, starting by trying to write a relatively gentler introductory section (to put immediately after the lead) to motivate and explain exponential functions in general and the "natural" exponential function in particular. What I have so far is:

Generalization of integer exponents

The most elementary description of an exponential function of base ⁠$b{\vphantom {)}}$⁠ is as the generalization of exponentiation to exponents which are arbitrary real numbers. For a positive integer exponent ⁠$n$⁠, exponentiation can be defined as repeated multiplication:

$$b^{n}=\underbrace {b\times b\times \cdots \times b} _{n{\text{ times}}}.$$

This definition establishes the essential property that addition of exponents is equivalent to multiplication of exponentials:

$$b^{m+n}~=~\underbrace {b\times \cdots \times b} _{m+n{\text{ times}}}~=~\underbrace {(b\times \cdots \times b)} _{m{\text{ times}}}\times \underbrace {(b\times \cdots \times b)} _{n{\text{ times}}}~=~b^{m}\times b^{n}.$$

Functions of the form ⁠$f(n)=A\times b^{n}$⁠ can be used to model phenomena which involve repeated multiplication. For example, imagine a simple model of a bacterial population, initially ⁠$1{,}000$⁠, which doubles every hour: each hour, the population will be multiplied by ⁠$2$⁠ compared to the population an hour before. After ⁠$t{\vphantom {)}}$⁠ hours, the population will have doubled ⁠$t{\vphantom {)}}$⁠ times, so will equal the initial population times an exponential of base ⁠$2$⁠: ⁠$P(t)=\textstyle 1{,}000\times 2^{t}$⁠. For instance, after ⁠$8$⁠ hours the population will be ⁠$P(8)=\textstyle 1{,}000\times 2^{8}=256{,}000$⁠.

But we might want to know the population after a fractional amount of time, such as after half an hour. If the amount of time is a rational number of hours, the definition of exponentiation can be extended using ⁠$n$⁠th roots, in a way that preserves the addition-to-multiplication property described above: ⁠$\textstyle b^{k/n}={\bigl (}{\sqrt[{n}]{b}}~\!{\bigr )}{\vphantom {)}}^{k}={\sqrt[{n}]{b^{k}}}$⁠. In the example, the bacterial population after half an hour will be ⁠$\textstyle P{\bigl (}{\tfrac {1}{2}}{\bigr )}=1{,}000\times {\sqrt {2}}\approx 1{,}414$⁠.

Quantities which change by repeated multiplication have a second essential property: their rate of change is proportional to their value. In the bacteria example, the population change ⁠$\Delta P$⁠ over a time increment ⁠$\Delta t$⁠ is the difference of the population before ⁠$P(t)$⁠ and the population after ⁠$P(t+\Delta t)$⁠, and the population after can be found by multiplying the initial population by a power of ⁠$2$⁠:

$${\begin{aligned}\Delta P&=P(t+\Delta t)-P(t)\\[3mu]&=P(t)\times 2^{\Delta t}-P(t)=\left(2^{\Delta t}-1\right)P(t).\end{aligned}}$$

So the rate of change of population is proportional to the current population, with a proportionality constant depending only on the time increment:

$${\frac {\Delta P}{\Delta t}}={\frac {2^{\Delta t}-1}{\Delta t}}P(t).$$

For irrational values of ⁠$x$⁠, a definition of the exponential ⁠$b^{x}$⁠ based on integer powers and roots alone is not sufficient, but the definition above for rational exponents can be extended to a continuous function of an arbitrary real-number exponent using the calculus concept of a limit. There are several ways to do this formally, but the most convenient ones come from describing the exponential of any base in terms of the so-called natural exponential function ⁠$\textstyle \exp(x)=e^{x}$⁠ with the "natural base" ⁠$e$⁠⁠${}\approx 2.718$⁠.

For any two real numbers ⁠$a$⁠ and ⁠$b{\vphantom {)}}$⁠, an exponential of base ⁠$a$⁠ can be written as an exponential of base ⁠$b{\vphantom {)}}$⁠:

$$a^{x}=\left(b^{\log _{b}a}\right)^{x}=b^{(\log _{b}a)x}.$$

This works because ⁠$\log _{b}$⁠, the logarithm of base ⁠$b{\vphantom {)}}$⁠, is the inverse of the exponential of base ⁠$b{\vphantom {)}}$⁠, which always satisfies ⁠$\textstyle b^{\log _{b}a}=a$⁠. In particular, any exponential can be written as an exponential of base ⁠$e$⁠:

$$b^{x}=\left(e^{\ln b}\right)^{x}=e^{(\ln b)x}=\exp(x\ln b),$$

where ⁠$\ln$⁠ is the natural logarithm, short for ⁠$\log _{e}$⁠. This identity can be used as a definition of the exponential function of base ⁠$b{\vphantom {)}}$⁠, assuming the ⁠$\exp$⁠ and ⁠$\ln$⁠ functions have already been defined.

The exponential function ⁠$\textstyle \exp(x)=e^{x}$⁠ with base ⁠$e$⁠ is the unique exponential function such that the derivative (instantaneous rate of change, the limit of the rate of change for infinitesimally small increments of the variable) is not merely proportional but is exactly equal to its value, with proportionality constant ⁠$1$⁠:

$${\frac {d}{dx}}e^{x}=\lim _{\Delta x\to 0}{\frac {e^{\Delta x}-1}{\Delta x}}e^{x}=1\times e^{x},$$

so ⁠$e$⁠ satisfies:

$$\lim _{\Delta x\to 0}{\frac {e^{\Delta x}-1}{\Delta x}}=1.$$

An exponential function of base ⁠$b{\vphantom {)}}$⁠ turns out to have a derivative equal to the natural logarithm of ⁠$b{\vphantom {)}}$⁠ times its value,

$${\frac {d}{dx}}b^{x}=\ln b\times b^{x}.$$

I'm aiming for something that would be at least moderately accessible to high school students encountering exponential functions for the first time who haven't learned much about calculus before, while trying not to get excessively long or let the topic scope drift too far. Not sure how well I'm threading that needle. @Hesselp Would something like that that address your concerns about making sure to frame our discussion relative to arbitrary bases, etc.? @D.Lazard, does this seem concise enough or is it too discursive / unfocused for an introduction here? –jacobolus (t) 21:33, 22 November 2024 (UTC)

@Jacobolus. Not a good adviser here, but can you just remove the calculus properties instead? I mean, if you are targeting the high school students, I think the relationship between logarithm and root may be moderately accesible for the beginners. Calculus can be put alongside describing the graph of its function. Dedhert.Jr (talk) 01:00, 23 November 2024 (UTC)

@Dedhert.Jr – This whole topic is ultimately dependent on calculus, which is therefore unavoidable. What I mean when I say I want to make it moderately accessible to someone who does not yet know calculus is that I don't want such a reader to immediately hit a wall and be left with no clue what is going on. If they can make it halfway through the first section after the lead and then we give them bit of in-line gloss explaining what a derivative is, they might be able to at least get the gist, or have some idea where to look next (e.g. by clicking through wikilinks to Calculus, Limit (mathematics), Derivative, Continuous function, etc., all of which should hopefully also be moderately accessible). (But it's also possible that the description could be more accessible than what I wrote here, or that I'm overdoing it and belaboring this section, with the narrative left clearer by moving or trimming parts.... feedback is welcome.) –jacobolus (t) 01:24, 23 November 2024 (UTC)

@Jacobolus Hmm? Really? Mine was never taught about the relationship between calculus and exponential functions. The integral and derivative were meant to be studied in the higher classes. (I have forgotten whether they studied these, so I might need to check again). Am not sure how U.S. classes study with these complexities, especially since I have never lived in the U.S. before. Dedhert.Jr (talk) 04:47, 23 November 2024 (UTC)

A typical US "precalculus" textbook (I just skimmed about 4) doesn't really say much about exponential functions. The books I looked at note that they can take a form like ⁠$f(x)=a^{x}$⁠, show pictures of graphs of such functions and describe the basic features of those graphs (e.g. the domain and codomain, the ⁠$y$⁠-intercept, the ⁠$x$⁠-axis as a horizontal asymptote), give a few examples of exponential growth and decay, discuss compound interest a bit, then give some problems to practice manipulating formulas containing logarithms and exponents or using a calculator to calculate related quantities. There's a mention in passing that there's a special example with base ⁠$e=2.71828...$⁠, which is not more carefully defined or described, and maybe some examples about the normal distribution.

In general the topic is deferred to calculus courses and beyond for more in-depth coverage. –jacobolus (t) 05:48, 23 November 2024 (UTC)

This draft is much better than the present section § Overview, and seems to have the same purpose. So, I would agree to place the draft just after the lead, and to remove § Overview, after having moved elsewhere parts and citations of § Overview that are not already present (as I just did for section "Formal definition").

However, two remarks: Firstly, the heading could be "Motivation". Secondly, this draft is intended for beginners. I suppose that such readers would have difficulties to read and understand the complicate formulas in which ⁠$\Delta$⁠ occurs. Therefore, we must think about replacing these formulas with prose. D.Lazard (talk) 16:18, 23 November 2024 (UTC)

Let me work on these as a draft for a while yet before trying to incorporate it in the article. You are probably right that this is too equation heavy; I'll try to figure out how to lighten that part. –jacobolus (t) 17:06, 23 November 2024 (UTC)

@D.Lazard: how about this version?

Motivation: exponentiation as a continuous function

An exponential function of base ⁠$b{\vphantom {)}}$⁠ is a continuous function ⁠$f(x)=b^{x}$⁠ whose value is the base raised to the power of the variable ⁠$x$⁠. Such functions generalize exponentiation with integer exponents to arbitrary real-number exponents.

For a positive integer exponent ⁠$n$⁠, exponentiation has an elementary definition as repeated multiplication,

$$b^{n}=\underbrace {b\times b\times \cdots \times b} _{n{\text{ times}}}.$$

Functions of the form ⁠$f(x)=A\times b^{x}$⁠ can thus be used to model phenomena which involve repeated multiplication, which appear ubiquitously in mathematics and science (see § Exponential growth). For example, imagine a simple model of a bacterial population, initially ⁠$1{,}000$⁠, which doubles every hour: each hour, the population will be multiplied by ⁠$2$⁠ compared to the population an hour before. After ⁠$t{\vphantom {)}}$⁠ hours, the population will have doubled ⁠$t{\vphantom {)}}$⁠ times, so will equal the initial population times an exponential of base ⁠$2$⁠: ⁠$P(t)=\textstyle 1{,}000\times 2^{t}$⁠. For instance, after ⁠$8$⁠ hours the population will be ⁠$P(8)=\textstyle 1{,}000\times 2^{8}=256{,}000$⁠. However the above definition only supports calculating the population once per hour; to find the population at other times requires further development.

The definition of exponentiation as repeated multiplication establishes the basic rule that addition of exponents is equivalent to multiplication of exponentials,

$$b^{m~\!+~\!n}~=~\underbrace {b\times \cdots \times b} _{m+n{\text{ times}}}~=~\underbrace {(b\times \cdots \times b)} _{m{\text{ times}}}\times \underbrace {(b\times \cdots \times b)} _{n{\text{ times}}}~=~b^{m}\times b^{n},$$

leads to the other rules of exponents,

$$b^{0}=1,\quad b^{-n}={\frac {1}{b^{n}}},\quad b^{~\!m~\!-~\!n}={\frac {b^{m}}{b^{n}}},\quad b^{m~\!\times ~\!n}={\bigl (}b^{m}{\bigr )}^{n}={\bigl (}b^{n}{\bigr )}^{m},$$

and, to maintain consistency, suggests a natural definition of ⁠$\textstyle b^{1/n}$⁠ as the ⁠$n$⁠th root of ⁠$b{\vphantom {)}}$⁠, so that a rational exponent can be expanded as

$${\begin{aligned}b^{k/n}&={\bigl (}b^{k}{\bigr )}^{1/n}={\sqrt[{n}]{b^{k}}}\\&={\bigl (}b^{1/n}{\bigr )}^{k}={\bigl (}{\sqrt[{n}]{b}}~\!{\bigr )}{\vphantom {)}}^{k}.\end{aligned}}$$

In the bacterial population example, if the amount of time is a rational number of hours, this extended concept of exponents can be used. For instance, the population after half an hour will be ⁠$\textstyle P{\bigl (}{\tfrac {1}{2}}{\bigr )}=1{,}000\times {\sqrt {2}}\approx 1{,}414$⁠. However, this is still not enough to answer questions about the model based on the concepts and tools of calculus, such as the population's derivative (instantaneous rate of change).

For arbitrary real-number values of ⁠$x$⁠ (which might be irrational), the definition of the exponential ⁠$\textstyle b^{x}$⁠ based on integer powers and roots alone is not sufficient, but it can be extended to define a continuous function using the calculus concept of a limit. The most immediate way to do this is to find a convergent sequence ⁠$\textstyle \left(q_{1},q_{2},q_{3},\ldots \right)$⁠ of rational exponents whose limit is ⁠$x$⁠, and define the exponential ⁠$b^{x}$⁠ as the limit of the exponentials ⁠$\textstyle \left(b^{q_{1}}\!,\,b^{q_{2}}\!,\,b^{q_{3}}\!,\,\ldots \right)$⁠. However, this definition is inconvenient, unenlightening, and cumbersome to work with, so instead the exponential of any base is usually defined in terms of the so-called natural exponential function ⁠$\textstyle \exp x=e^{x}$⁠ with the "natural base" ⁠$e$⁠⁠${}\approx 2.718$⁠, the unique exponential function whose derivative is equal to its value, ⁠$\textstyle {\tfrac {d}{dx}}\exp x=\exp x$⁠.

For any two real numbers ⁠$a$⁠ and ⁠$b{\vphantom {)}}$⁠, an exponential of base ⁠$a$⁠ can be written as an exponential of base ⁠$b{\vphantom {)}}$⁠:

$$a^{x}=\left(b^{\log _{b}a}\right)^{x}=b^{(\log _{b}a)x}\!.$$

This works because ⁠$\log _{b}$⁠, the logarithm of base ⁠$b{\vphantom {)}}$⁠, is the inverse of the exponential of base ⁠$b{\vphantom {)}}$⁠, so it satisfies ⁠$\textstyle b^{\log _{b}a}=a$⁠. In particular, any exponential can be written as an exponential of base ⁠$e$⁠:

$$b^{x}=\left(e^{\log _{e}b}\right)^{x}=e^{(\log _{e}b)x}=\exp(x\ln b),$$

where ⁠$\ln$⁠ is the natural logarithm, short for ⁠$\log _{e}$⁠. This identity can be used as a definition of the exponential function of base ⁠$b{\vphantom {)}}$⁠, assuming the ⁠$\exp$⁠ and ⁠$\ln$⁠ functions have already been defined.

The exponential function of arbitrary base ⁠$b{\vphantom {)}}$⁠ turns out to be just the natural exponential function with a re-scaled domain. It is common to solve problems involving arbitrary exponentials by first rewriting them as exponentials of base ⁠$e$⁠, which makes them easier to combine, compare, and analyze by simplifying the relevant identities and reducing the number to remember. Instead of naming the base ⁠$b{\vphantom {)}}$⁠, it is common to name its natural logarithm, e.g. ⁠$k=\ln b$⁠, and write the generic ⁠$b^{x}$⁠ as ⁠$e^{kx}$⁠ or ⁠$\exp kx$⁠ instead.

For example, the derivative of an arbitrary exponential function can be found by applying the chain rule,

$${\frac {d}{dx}}e^{kx}=e^{kx}\cdot {\frac {d}{dx}}kx=ke^{kx}.$$

I rearranged things a bit and tried to take out some of the more technical bit, while also including more of the basic rules of exponentiation near the top. –jacobolus (t) 05:09, 24 November 2024 (UTC)

Fine. At first glance, I have no major remark. Maybe, "convenently analyzed as such" could be replaced by a sentence saying explicitly that the common way of solving problems involving exponentials of base ⁠$b$⁠ is to convert them to natural exponentials since this this make formula manipulation much simpler by limiting the number of formulas that have to be remembered. (You are definitely better than me for finding a good formulation). D.Lazard (talk) 09:49, 24 November 2024 (UTC)

How's this update? –jacobolus (t) 16:27, 24 November 2024 (UTC)

@Jacobolus:   Your title (Motivation: exponentiation . . .) announces a (sub-)section about the two-arguments function called 'exponentiation'. So your text should be in the existing article 'Exponentiation'. In fact that's already the case, see its lead and section 'Real exponents'.
Why do you want to duplicate it in an article about the one-argument functions, with as primary topic . . . the exponential function ⁠$\exp$ (Jacobolus 23 Nov.) ?   The function exp can be defined in more ways than using exponentation: by series, by the limit of compound interest and by continued fractions.  All four should be mentioned, but without duplicating the specialized articles. In my opinion.   Hesselp (talk) 17:41, 2 December 2024 (UTC)

My main motivation is trying to compromise with you / address your complaints without the significant topic shift you are otherwise proposing (which I am opposed to and think is unlikely to find consensus). At the same time I think it's helpful to give a gentler introductory section to readers (e.g. high school students) who don't know any calculus but encounter the topic. As you say, this is somewhat duplicated material summarizing exponentiation: that's the point. –jacobolus (t) 18:20, 2 December 2024 (UTC)

I don't resist (anymore) the idea that the main issue in this article will be the function with symbol exp. Provided that there will be one section on its 'generalizations' (as announced in the hatnote). Hesselp (talk) 20:48, 2 December 2024 (UTC)

Section "Computation of a^b where both a and b are complex"

Section "Computation of a^b where both a and b are complex" is a mess. A minor issue is that its heading cannot be linked correctly, because it uses templates. A major issue is that its content belongs to exponention rather than this article. Moreover the section is so poorly written that it is almost not understandable. Also, it is mathematically contovesial. For example, it qualifies as singularities the points that are not singular if the function is considered as a multivalued function and are singular only if the function is restricted to its principal value.

For these reasons, I'll remove the section. D.Lazard (talk) 09:48, 27 November 2024 (UTC)

On section 'General exponential functions'

Comments about the current version (4 Dec.)
Sentence 1
Again, what could be meant with 'generalizations of a given function' ?  Any sources?
Sentence 2
The meaning of "These functions" is unknown as long as the meaning of "generalizations" is unknown.
Functions don't have a (unique) form. But a well determined function can be represented by a number of more or less usual expressions.
Sentence 3
The meaning of "Such a function" is unknown as long as the meaning of "generalizations" is unknown.
Not only "such a function", but every function can be expressed in terms of the function $\exp$. See Talk 2 Dec. 17:11 .
Why $\alpha$ as parameter? In section 'Derivatives and differential equations' it is k. As well in the lead until 22 Nov. Letter $\alpha$ suggests a – not existing – relation with the x=0-value written as a.
Sentence 5
"These exponential functions" is not defined.
It isn't clear that $\alpha$ doesn't depend on x . Better: "The value of $\,f'(x)/f(x)\,$ doesn't depend on x.".
Sentence 6
This seems to be meant ("characterize") as a definition of 'generalized exponential function'. But in the definition you cannot speak of 'this property (of generalized exponential functions)'. Better: 'This condition . . .'.
The definition should be placed at the start, before the name 'generalized exponential function' is used.
Sentence 7
This is another wording for conditions 1 and 2 in the Article version 2 Dec. second sentence. Possibly this was considered as OR by Lazard in his Summary 3 Dec..
General
I agree to the changes from Quantling, 3 Dec.   Hesselp (talk) 22:39, 8 December 2024 (UTC)

@D.Lazard: I can partially agree with your changes/simplifications after 2 januari 2025. But, according to me:

• The name "exponential function" shouldn't be used before its meaning is introduced/defined. So the remarks in sentences 2-5 (Art. 18:12 5 Januari 2025) should belong to subsection 'Expression…', and the note to 'Hierarchy…' second point. Exponential functions of time (not unusual) cannot be written in the form $x\mapsto ab^{x}$.
  
• The defining condition "same differences, same ratios" is easiest and most well known to non-mathematicians: without derivation or exponentiation. So it should be mentioned on top.
  
• The double meaning of 'base' deserves more emphasis; an example with concrete numbers instead of variables will be more convincing.
  Hesselp (talk) 19:27, 5 January 2025 (UTC)

  I don't understand: you are saying that the phrase "exponential function" should not be used in a section entitled "General exponential function" placed after as section defining the (natural) exponential function. What is the rationale for that?

  About rational functions of the time: I guess that you mean functions of the form ⁠$t\mapsto y_{0}d^{k(t-t_{0})}.$⁠ If you define ⁠$b=d^{k}$⁠ and ⁠$a=y_{0}d^{-kt_{0}},$⁠ the function takes the form ⁠$t\mapsto ab^{t}$⁠. (Should I recall you that the change of the variable name does not change a function?) D.Lazard (talk) 10:22, 6 January 2025 (UTC)

@D.Lazard: Thanks for your explanations. The first sentence of the lead defines the function $exp$. And further on I see "functions of the general form ⁠$f(x)=b^{x}$⁠" and "functions of the form ⁠$y(t)=ae^{ct}$⁠". But I can't see a function (or another mathematical concept) as defined , by nothing more than mentioning the/a usual way of its written notation/form. That's my 'rationale' for starting the subsection "General exponential function" with an attempt to really define the phrase (the objects with the name) 'exponential function' in the extended sense.

How can I/someone decide whether or not the function $x\mapsto 2^{x^{2}}$ should be called an exponential function, using your criterium $=ab^{x}$ or $=ae^{kx}$ ?

My problem with the notation $x\mapsto ab^{t}$ for exponential functions of (the) time is that I can't interpret (and I suppose many readers can't either) an exponentiation with an exponent which is not dimensionless.

On "(the standard form of these functions must be given)" in your last summary. Several usual forms are mentioned in the subsection Expressions …', directly after the defining conditions.

Last point. According to me, "differentiable" in sentence 2 can be missed. For the occurrence of $f'$ in the second condition, together with 'equivalent' in sentence 2, implies already the differentiability. Hesselp (talk) 23:27, 6 January 2025 (UTC)

@D.Lazard: I understand and agree with your argument (showing expressions that beginners could have seen before). I interprete your 'characterizations' (I cannot remember I've ever met this word before) as meaning the same as 'expressions'.
But I should prefer to mention the usual expressions in one of the very first sentences of the article; the third section after the lead is rather late, I think. See my proposals 31 Oct. (third sentence) [[1]] and 6 Nov. (first sentence) [[2]]. For now I'll stay away from the (extremely long) intro; see the adaptation in the Article. Hesselp (talk) 13:04, 12 January 2025 (UTC)

@D.Lazard: About 'characterizations' I learned a lot the last days, starting with "a characterization of an object . . . is logically equivalent to [the definition of the object]" (first sentence in Characterization (mathematics)).  And in Characterizations of the exponential function I don't find a characterization of the natural exponential function resembling your proposals. So your two expressions (as you call them in your summary 12 Jan.) should not be placed after "the following equivalent conditions:". Because: see Talk 6 Jan. .
Your proposal can be seen as properties of exponential functions (except that $b^{x}$ is not defined for a non-numerical exponent), not generally applicable tests deciding whether or not a given function can be called exponential.   As an alternative I suggest to start the article with:
"Exponential functions are often represented by written forms as $x\mapsto ab^{x}$, $x\mapsto ae^{kx}$, $x\mapsto ae^{x/\tau }$ for some constants $a$, $b(>0)$, $k$, $\tau$.  The most important one in pure mathematics is called the exponential function or the natural exponential function, written as $x\mapsto e^{x}$ or $x\mapsto \exp(x)$. It is the unique real function . . ." . Hesselp (talk) 22:17, 14 January 2025 (UTC)

The beginning of the article is the result of a consensus between several editors that include competent mathematicians. Please stop trying to change it without strong reasons. In your long posts I do not see any strong reason for changing the first sentence, and I see several reasons for which your proposal in a disimprovement.

Also, it seems that you consider yourself as the owner of the article, since you consider other's edits as "proposal", that you revert simply because of your feeling of what the article should be. This is not a good practice; see WP:OWN.

Before your edits, the article was not very good and had many issues, but presently, it is much worse, despite the time I spent ubsuccesfully to fix your poor edits, and to convince you of your errors. I consider that most of your edits in the article and this talk page are WP:disruptive editing. It is for keeping time for improving other articles that, generally, I do answer your posts, and I fix your edits only when they are too confusing for readers. D.Lazard (talk) 04:04, 15 January 2025 (UTC)

@D.Lazard: Splitting up the article in a 'general' section, versus an extensive treatment of the function $exp$ in the rest, is proposed/discussed on this talk page (e.g. 22, 24 Nov. 2024), without clear objections. A consequence is, imo, that a list of more or less usual expressions for the 'general' case should be shown in this section anyway, even if some of them (I found only one) occur in other places. Hesselp (talk) 23:45, 24 January 2025 (UTC)

@Hesselp after reverting your last edit, I came here to say that I concur with @D.Lazard: please stop making unnecessary changes to this article. Overall, your edits have decreased the quality of the article (and made other editors lose time).

If — as you say yourself — you have only recently encountered the word "characterization" for the first time, maybe that should make you ask yourself some questions? For instance, maybe English is not your native language, and you are not proficient enough in it to edit math articles on en.wp; or maybe you are not familiar enough with mathematical texts to be making constant changes to math articles on Wikipedia. Whatever the answer might be: please take into account what other editors have been telling you.

Cheers, Malparti (talk) 14:28, 22 January 2025 (UTC)

@Malparti: I took note of your remarks 22 Jan. Mathematics is a very comprehensive area of knowledge. I'm aware that I know little about the greater part of it.
About your revert. 'Common characterization' (the name used by D.Lazard) (iii) I see at least as common as (ii), so why not mention this type as well. Moreover, they both occur in (v). Numbering the different variants symplifies referencing (within this section and in Talk).
I chosed for letters u, v in some formulas; not x, y because y is also usual in y-axis and y(x). The x and y will be an unintended mistake.
I see the two sentences "Without using..." and "Providing the values..." as noteworthy, emphazising the differences in the way of characterization between the given defining conditions. (More or less comparable with the 'prose' mentioned by D.Lazard in his summary 22 Jan.?) . Can you explain why you consider the version without this sentences as preferable? I wait with a possible revert on this point. Hesselp (talk) 23:50, 24 January 2025 (UTC)

The two "sentences" removed by Malparti are not gramatically correct complete sentences. Moreover, they are confusing, since, being inserted between items of a list, the reader may not understand whether the following items belong, or not, to the same list. Finally, such emphasizing is clearly discouraged in Wikipedia; see MOS:NOTE.

"(iii) I see at least as common as (ii), so why not mention this type as well": This confirms your lack of competence in mathematics that you claim yourself. Indeed, every reader that is able to understand the other conditions should know that changing ⁠$k$⁠ into ⁠$1/\tau$⁠ or ⁠$3\alpha$⁠ does not changes the condition. Continuing this way would allow inserting hundreds of similar conditions. D.Lazard (talk) 09:45, 25 January 2025 (UTC)

Hi @Hesselp,

I think the following sentence in D.Lazard's answer is very important: "Continuing this way would allow inserting hundreds of similar conditions.". In mathematics — but that remark goes beyond mathematics — there are countless ways to say something, and one can always find additional things to say about a given topic. However, adding things has the effect of cluttering a text, which ultimately makes it less clear. And the goal of Wikipedia articles is to give a clear overview of everything that is important to know about a subject — not to collect every single piece of information on that subject. So, sometimes, less is more.

Finding the right level of detail for an article can be tricky, and often there are debates among editors about this. Most of the articles of articles on "basic" math topics — such as this article — have been read by a lot of people (more than one thousand people a day on average for "Exponential function"), and their current state is the result of a consensus between senior editors and recent modifications by editors with varying degrees of experience. By your own account, you have a limited knowledge of mathematics — like us all! But maybe more so than, e.g, D.Lazard — and no offense but you do not sound like a native speaker to me (if I had to guess, I would say you might be speaking a Germanic language?). There is nothing wrong with not knowing all about a topic (no one does) and not being a native speaker (I am not, and neither is D.Lazard — I think). But those are two things you have to take into account when editing an article.

In sum, although I know your intentions are good, I think you should step back a bit and ask yourself: "Why do I want to keep editing that article so much, when several other editors have been telling me what I am doing is not helping?". If the answer is that if you are convinced that your edits are improving the article, then unfortunately you seem to be the only one to think so. If it is that you want to contribute to Wikipedia: that's great, and there are many ways to do so. I'm sure your specific set of skills will be super useful for some articles. My advice would be to be humble at first and start with small "beginner" tasks and then as you learn what Wikipedia is and how it works, you can move up to bigger edits — but that's just my advice; I have no authority to tell you what you should be doing on Wikipedia. ;)

Best, Malparti (talk) 11:41, 25 January 2025 (UTC)

"current state is the result of a consensus between senior editors and recent modifications by editors with varying degrees of experience" – to be fair though, often, as in this example, the current state is pretty mediocre and could be significantly improved with more work. –jacobolus (t) 04:40, 26 January 2025 (UTC)

@Jacobolus I agree! I realize now that I read it back that my comment gives the impression that I am very "anti change by new editors"... Which isn't at all the case. What I actually think is closer to "when it comes basic math articles, more articles would benefit from being streamlined than from the addition of new information". Malparti (talk) 10:59, 26 January 2025 (UTC)

Section "Expressions for exponential functions"

A user restored the previously deleted section § Expressions for exponential functions. I'll remove this section again for the following reasons

• It does not respect MOS:PROSE
• It uses ⁠$\equiv$⁠ in a unusual and undefined sense
• All listed formulas are given earlier in the article or can be deduced from previously given formulas with easy substitutions.

Please, do not add this mess again. D.Lazard (talk) 10:06, 25 January 2025 (UTC)