Talk:Natural logarithm

Absolute values in logarithms

(Moved the following unsigned comment by User:David phys davalos from my talk page).--Sapphorain (talk) 00:10, 9 July 2021 (UTC)

The antiderivative of 1/x is not log(abs(x)), is simply log(x), please do not undo my changes. (Unsigned comment by User:David phys davalos)

Both ln|x| and ln(x) are antiderivatives of 1/x for x real and nonzero. But as at this point of the article the function ln(x) is only defined as a real function of the real positive variable x, we better stick to the former.--Sapphorain (talk) 00:10, 9 July 2021 (UTC)

Observe that if you are taking only x>0, abs(x) is precisely not needed, it introduces confusion since log(x) is in general different from log(abs(x)). I agree that they are the same for x>0, therefore you do not need the absolute value. The confusion is introduced because log(|x|) can be trivially extended for all real x (and real-valued), while log(x) does not enjoy such property. Moreover, is log(x) what is eventually extended, not log(abs(x)). In summary, as you said abs(x) does nothing, so let's make things simpler. — Preceding unsigned comment added by David phys davalos (talk • contribs) 20:41, 9 July 2021 (UTC)

In fact, observe that using the uneded abs(x) introduces inconsistencies when dealing with trigonometric functions, giving rise to functions defined in domains that they shouldn't be. I encourage you to leave things simpler (and in fact, correct).

I reverted you. Things are not simpler when they involve undefined notions, they are just not anymore understandable. --Sapphorain (talk) 21:27, 9 July 2021 (UTC)

Wow, wait, what notion is not defined?. You are making things involved since you are confusing domain issues with the function itself. Please provide a reference of your highly non-standard and non sense rules. — Preceding unsigned comment added by David phys davalos (talk • contribs) 23:18, 9 July 2021 (UTC)

If you do not want to 'believe' me, believe Wolfram: https://www.wolframalpha.com/input/?i=int%281%2Fx%2Cx%29 — Preceding unsigned comment added by David phys davalos (talk • contribs) 23:22, 9 July 2021 (UTC)

And: https://www.wolframalpha.com/input/?i=int%281%2Fabs%28x%29%2Cx%29

Now, please, let's stop this annoying game. You already said that for x>0 abs(x) does nothing, and the article is about x>0, abs(x) is totally not needed! and introduces confusion for the reader that eventually reads complex logarithms. — Preceding unsigned comment added by David phys davalos (talk • contribs) 23:26, 9 July 2021 (UTC)

At this point of the article, $ln(y)$ is defined exclusively for a real positive argument, as a real function. For a value of $x$ such that $y=\sec(x)$ is negative, for instance, $ln(y)$ is undefined.--Sapphorain (talk) 06:22, 11 July 2021 (UTC)

Confusion

As a non-mathematician, but engineer, this article is somewhat confusing. It indiscriminately mixes chat about any base logarithms in with that of the natural base logs, instead of focusing on the natural logs and how they are different from the others. It's missing what I recall as the defining characteristic of the natural logs: "But the one property that goes to the essence of e and makes it so natural for logarithms and situations of exponential growth and decay is this: d(e^x/dx) = e^x." 74.127.200.33 (talk) 18:45, 11 October 2022 (UTC)

Absent Axes

The graph on the right illustrating the curve has two axes. It is conventional and important for clarity to label axes. The axes on this graph are unlabelled, making it confusing to the non expert reader. The expert reader does not need to read this article, so it is necessary to clarify thing like this. — Preceding unsigned comment added by 209.93.146.80 (talk) 20:26, 4 February 2023 (UTC)

the discussion on units needs to be returned

It is false to say that the argument of a logarithm must be unitless. There is no mathematics to support this assertion. Engineers, physicist and chemists routinely take the logarithm of quantities with units and that most certainly does not violate any mathematical concepts. The comment:

this is false, that integral only works because it's a definite integral evaluating to log(b/a), i.e. we still cancelled units to make the inside unitless

Is irrelevant and false. ALL NATURAL LOGARITHMS ARE DEFINITE INTEGRALS. There is no such thing as a natural log that is not a definite integral. The divisor of a (to remove the units) in the post text above is incorrect and unneeded. The mathematical PROOF I provided shows that units are allowed in the argument of log function and that the resulting value is always unitless. Unless an editor can provide a mathematical PROOF that it is not true then the section on units should be returned.

here is the removed text:

Units

An often asked question is "what happens if the argument to the log function (natural or any other base) has units"? A Google search will likely provide the incorrect answer and return many pages that say the argument to a log function must be unitless. This is incorrect. The argument can, and often does, have units. However, the value returned by a log function is always unitless regardless of the units of the argument.

The simplest way to understand how the log function handles units is by examining the definition for the natural log as an Integral. The natural log of $x$ is the area under the curve of $x$ plotted against $1/x$ in the region from 1 to $x$. Note that the units of an integral are always the units of the x-axis times the units of the y-axis. But since for the logarithm the y-axis is defined as $1/x$, the resulting units of the area are $(units\ of\ x)/(units\ of\ x)$ and the units always cancel and the area is unitless. So the natural logarithm of a quantity with units is always unitless regardless of the unit of the quantity. Furthermore, there are no restrictions on the units of the argument, it can be unitless or have any unit desired.

It does not matter which mathematical definition is used for the natural log since all definitions are mathematically equivalent. Furthermore, in mathematics, units are treated exactly the same way the numeric values are. A mathematical equation must work with units or the mathematics is wrong.

A corollary of the way a logarithm handles units is that the log function is a "lossy function" and the inverse function can not return the original quantity. This is similar to the way squaring works. The inverse function, square root, can not return the original value of the function since squaring a number removes the sign, and the square root can not recover that lost sign. So the ${\textstyle \surd {x^{2}}<>x}$ for all $x$. For a logarithm, the exponent; $\exp(log(x))<>x$ for all $x$ since the log function has removed the units of $x$ and exponentiation can not recover the lost units. For example, $\exp(log(10\ meters))=10<>10\ meters$.