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636:(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.
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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
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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
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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
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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
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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:
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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.
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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).
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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
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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.
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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
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It does not matter which mathematical definition is used for the natural log since all definitions are mathematically equivalent. Furthermore, in mathematics,
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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: -->
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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.
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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.
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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
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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
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I reverted you. Things are not simpler when they involve undefined notions, they are just not anymore understandable. --
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The antiderivative of 1/x is not log(abs(x)), is simply log(x), please do not undo my changes. (Unsigned comment by
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0, abs(x) is totally not needed! and introduces confusion for the reader that eventually reads complex logarithms.
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and makes it so natural for logarithms and situations of exponential growth and decay is this: d(e^x/dx) = e^x."
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is defined exclusively for a real positive argument, as a real function. For a value of
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a number removes the sign, and the square root can not recover that lost sign. So the
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characteristic of the natural logs: "But the one property that goes to the essence of
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evaluating to log(b/a), i.e. we still cancelled units to make the inside unitless
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Now, please, let's stop this annoying game. You already said that for x: -->
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A mathematical equation must work with units or the mathematics is wrong.
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this is false, that integral only works because it's a definite integral
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https://www.wolframalpha.com/input/?i=int%281%2Fabs%28x%29%2Cx%29
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and exponentiation can not recover the lost units. For example,
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units are treated exactly the same way the numeric values are.
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https://www.wolframalpha.com/input/?i=int%281%2Fx%2Cx%29
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0 abs(x) does nothing, and the article is about x: -->
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If you do not want to 'believe' me, believe
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861:{\displaystyle (units\ of\ x)/(units\ of\ x)}
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