Talk:Logarithm

Exponentiation's other inverse function?

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→b^x (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 ⁠$\ln x$⁠ or ⁠$\log _{b}a$⁠. –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 $\exp _{b}(x)=b^{x}$ has an inverse logarithm function $\log _{b}(x)$. 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 $f(x,y)=x^{y}$ 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 $x,y$) 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 ⁠$y=b^{x}$⁠ the quantity ⁠$b$⁠ is called the base, the quantity ⁠$x$⁠ is called the exponent, and the quantity ⁠$y$⁠ is called the result. When this equation is rewritten to isolate the exponent ⁠$x$⁠, ⁠$x$⁠ is instead called the logarithm base ⁠$b$⁠ of ⁠$y$⁠, denoted ⁠$x=\log _{b}y$⁠." 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)

"it does not describe what the function is" ??

@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   f^inv(u) + f^inv(v) = f^inv(uv). f^inv(u+v) = f^inv(u) · f^inv(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 ⁠$\mathbb {R}$⁠ 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)

Definition of logarithm

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 $\log _{b}x$ is defined as $\ln b/\ln x$, where $\ln x$ is defined as an integral (or in some other equivalent way). The problem with the definition "The real $y$ such that $b^{y}=x$" is that the exponential $b^{y}$ is usually defined as $\exp(y\ln b)$, 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)

There are four choices for defining logarithm:

1) Euler's exponentials and inversion, the standard approach since 1748. This is a circular definition since exponentials are also transcendental functions. Flipping between b^y = x and log_b x = y gives the illusion of algebra without algebraic functions.

2) The Hyperbolic logarithm of St-Vincent and de Sarasa was used from 1647 to 1748; it relies on signed areas for definition.

3) Alfréd Haar proved in 1933 that every locally compact group has an invariant measure, as mentioned in the list of examples at Haar measure. The group mentioned is positive real numbers which is isomorphic to the group of squeeze mappings.

4) Since natural logarithm is nearly the same as hyperbolic angle, the angle can be explained from first principles and then the logarithm defined in terms of the angle. See v:Reciprocal Eigenvalues. — Preceding unsigned comment added by Rgdboer (talk • contribs) 22:12, 30 September 2024 (UTC)

Submitted in the spirit of WP:TALK#PROPOSE — Rgdboer (talk) 01:06, 1 October 2024 (UTC)

In the spirit of PROPOSE, what is your proposal? Dondervogel 2 (talk) 07:06, 1 October 2024 (UTC)

Approximation for splitting the logarithm of a sum

I think it could be interesting to add an approximation for splitting the logarithm of a sum which rises from approximating the Softplus function:

$\ln \left(1+e^{x}\right)\approx {\begin{cases}\ln 2,&x=0,\\[6pt]{\frac {x}{1-e^{-x/\ln 2}}},&x\neq 0\end{cases}}$

which leads to the following smooth approximation to the maximum function and LogSumExp function (not included in the smooth maximum page): with $x>0,\ y>0,\ x\neq y$

${\text{max}}\{x,\ y\}\approx \ln(e^{x}+e^{y})\approx {\dfrac {x\,e^{\frac {x}{\ln(2)}}-y\,e^{\frac {y}{\ln(2)}}}{e^{\frac {x}{\ln(2)}}-e^{\frac {y}{\ln(2)}}}}$

with just a change of variables it will lead to how approximate splitting the sum of a logarithm as:

$\ln(x+y)\approx {\dfrac {\ln(x)\,x^{\frac {1}{\ln(2)}}-\ln(y)\,y^{\frac {1}{\ln(2)}}}{x^{\frac {1}{\ln(2)}}-y^{\frac {1}{\ln(2)}}}}$

I believe is interesting since I have never seen before the approximation for splitting the logarith of a sum before finding it accidentally here {https://math.stackexchange.com/q/4838311/909869}. Maybe a superuser could incorporate it after checking it do works as intended. 45.181.122.234 (talk) 02:24, 25 September 2024 (UTC)

Essential definition and explanation

"[...]the logarithm of a number x to the base b is the exponent to which b must be raised to produce x." Avoiding passive voice and weak words forces writers to drive sentence flow comprehensively.

(Avoid words similar to "be", "is", "are", "will", "was", "am", as well as "as", "become", "do", "make", "have", "with", "use", "can", "cannot", "through", "very", "more", "wrong", "by", "cause", "result", "surmount", "possess", "comprise", "consist", "correlate", "link", "create", "because" etc. This forces writers into chronological and detailed sentence flow.)

I suggest strengthening the opening explanation and definition of a logarithm by including the following information:

[logb^x = y; b^y = x]; [logb^(b^x) = x]:

[logb^x = y; b^y = x]: log base exponent results in a new exponent of the base that results in the original exponent. [log] causes [b^x] to equal [y] and [b^y] to equal [x]

[(logb^y = x; b^x = y) = (logb^x = y; b^y = x)] exponent formula [b^x = y] represents an inversive function of [logb^y = x] log formula [logb^x = y] represents an inversive function of [b^y = x]

[logb^(b^x) = x]: [(b^x = y) = (log b^(b^x))]

This better explains the characteristics of the logarithm formula through algebraic relationships which novices, (me), seek from the top of the article. 2600:1700:3830:4820:44C0:78D3:2AAE:53CE (talk) 11:52, 11 November 2024 (UTC)

Instead of teaching how to write mathematics, you should better propose a better formulation of the leading sentences. Also the information you suggest to add seems to be already present in the article, especilly in section § Logarithmic identities. I wrote "seems", because your formulas do not use any common notation, and your prose use formulations that are neve defined (for example, what is a "log base exponent"?) This makes your suggestions totally ununderstandable. D.Lazard (talk) 12:15, 11 November 2024 (UTC)

In Wikipedia, it is essentially to tell the audience who are not good at mathematics to understand what logarithm is all about. Putting the unfamiliar symbols and mathematical notations and replacing articles, verbs, and other vocabularies with them initially is why the audience failed to understand it: it is difficult. Dedhert.Jr (talk) 13:21, 11 November 2024 (UTC)

Adding an interactive example to the Examples section

User:ZergTwo recently reverted my edit [1], and I thought I'd ask for a second opinion.

log ₁₀ ( 100 ) = 2

Interactive logarithm calculator

I think it would be interesting to add an interactive log calculator to the examples or definition section (i.e. Like the box on the right). This could serve to illustrate the definition of the log() function in a very similar way to how a plot of the log function does. Like a plot, it would illustrate how the result of the log function varies as you vary its input or its base, but presented in an interactive way instead of a visual way. The reader can adjust the input number and see how that affects the output number. I know this is a little different from an image, but I think it would be just another way to illustrate the topic of the article. Bawolff (talk) 07:25, 10 June 2025 (UTC)