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I know that :

$$\ln: (0,+\infty )\to \mathbb{R}$$ $$\ln(x):=\int_1^x \dfrac{dt}{t}$$

And :

$$\ln^{-1}: \mathbb{R} \to \mathbb{R}^+$$ $$\ln^{-1}(x):=\exp(x)$$

after that. We define :

$$f: \mathbb{R} \to \mathbb{R}^+$$ $$f(x):=\big(\exp(\ln a)\big)^x \ \ \ a >0 ,a \neq 1$$

And :

$$f^{-1}: \mathbb{R}^+ \to \mathbb{R}$$ $$f^{-1}(x):=\log_a (x) \ \ \ a >0 ,a \neq 1$$

now my question : function $\dfrac{1}{t}$ How did they get?

Martin Sleziak
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Almot1960
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    Why is "history" in your title? – Dave L. Renfro Jun 13 '17 at 20:30
  • @DaveL.Renfro.For this purpose: How was the logarithm discovered? – Almot1960 Jun 13 '17 at 20:33
  • See this 1913 paper by Florian Cajori. Note that this is the last part of a mult-part paper. Look at the end of this paper to see the previous parts of this multi-part paper. – Dave L. Renfro Jun 13 '17 at 20:33
  • Another useful reference is Glasher's 1920 paper On early tables of logarithms and the early history of logarithms, but you might have to visit a university library (or have online access to one), since I wasn't able to find it freely available on the internet. – Dave L. Renfro Jun 13 '17 at 20:40
  • @DaveL.Renfro. ok thank you . – Almot1960 Jun 13 '17 at 20:42
  • For a more popular account, you might look at "e: The Story of a Number" – Robert Israel Jun 13 '17 at 21:38
  • I don't know the true history, but it seems very natural to first define $b^x$ and then define $\log_b(y)$ to be the solution to $b^x = y$. The number $e$ is discovered as the special value of $b$ that makes the derivative of $b^x$ equal to $b^x$. It seems to me that defining $\ln(x)$ to be $\int_1^x \frac{1}{t} , dt$ is something someone would only think of as an afterthought, when they are searching for the most efficient way to define these functions and prove their properties rigorously (for example if they are writing a real analysis textbook). – littleO Oct 23 '20 at 03:07

2 Answers2

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The discovery of (decimal) logarithms by Napier (tables published in 1614) predates by a century or so the discovery of the fact that the area between $1$ and $a$ under the hyperbola with equation $y=\dfrac{1}{x}$ has a value which is proportional to a multiple of the (decimal) logarithm of $a$. It was not until the second half of the 18th century, with the fixation of notations by Euler, who in particular introduced the natural logarithm (also called Euler logarithm, now denoted $\ln$) that it became current to write

$$\int_a^b \dfrac{dx}{x}=\ln(b)-\ln(a)$$

(the integral notation was introduced only 50 years earlier, by Leibnitz).

It was the same Euler who fixed the notation $e^x$ for the exponential function, and explained in a plain way that exponential and logarithm are inverse functions one of the other.

Jean Marie
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The May 2022 issue of The College Mathematics Journal will contain the article "The Equivalence of Definitions of the Natural Logarithm Function," which has some historical comments and a 25-item bibliography.

PolyaPal
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