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The natural logarithm function ($\ln x$) and the base of the natural logarithm function ($e$) are both extremely useful. They're also both closely related: $\ln (e^x)=x$, and $e^{\ln x}=x$. But which came first? I would think it's likely they were developed together, but each could have been developed separately. For example, $\int 1/x \,dx=\ln x$, and the $\cosh$ function can be described in terms of $e$. So which came first: the natural logarithm function or the base of the natural logarithm function?

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  • $\begingroup$ James Whitbread Lee Glaisher's historical survey paper On early tables of logarithms and the early history of logarithms [Quarterly Journal of Mathematics (Oxford) (1) 48 (1920), 151-192] is very informative, but it doesn't seem to be freely available on the internet. $\endgroup$ – Dave L Renfro Nov 20 '14 at 21:41
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It may seem strange but logarithms were invented much earlier. Napier used the base $(1-10^{-7})^{10^7}$ which is very close to 1/$e$ (within 0.00000002 of 1/$e$). Number $e$ (as a limit) was formally defined by Euler approximately 100 years after Napier.

Napier's MIRIFICI LOGARITHMORUM CANONIS CONSTRUCTIO (English translation by Ian Bruce) contains tables of logarithms, and explanations of the constructions of the tables.

EDIT. Natural Logarithms and the formula $\ln x=\int_1^xdx/x$ defining them were known long before Euler. Modern texts usually define them as the inverse function of $e^x$ but historically this was not the case: $e^x$ is much later invention than logarithms. According to Wikipedia, this definition using "the area under the hyperbola" is due to Alphonse Antonio de Sarasa (1649), that is a century before Euler.

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    $\begingroup$ I'll upvote this - if you can add a source. $\endgroup$ – HDE 226868 Oct 29 '14 at 2:03
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    $\begingroup$ Any history of math. You may begin with Wikipedia. $\endgroup$ – Alexandre Eremenko Oct 29 '14 at 2:05
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    $\begingroup$ You can look at the original (in translation): 17centurymaths.com/contents/napier/constructiobookone.pdf $\endgroup$ – Will Orrick Oct 31 '14 at 2:21
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    $\begingroup$ Good answer, i upvoted it. But you should add a sentence that actually answers the OP's question. He asked specifically about the natural logarithm 'ln'...so what i gather from your question is basically, that logarithms in general were already known, and a numerical approximation of e was already known, but until Euler established e as a limit the 'real' natural logarithm wasn't invented? So e and ln were born simultaniously? $\endgroup$ – Matthaeus Nov 20 '14 at 14:40
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    $\begingroup$ @Matthaeus: Napier's logarithms were not natural and were not logarithms, strictly speaking. But that his base was close to $e$ shows that he somehow understood what the "natural logarithms" and the "natural base" is. $\endgroup$ – Alexandre Eremenko Nov 20 '14 at 20:11
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Logarithm tables have been used since at least the middle ages by merchants in order to perform big multiplications. Guess that makes them come first, although the formal definition came latter, as shown by Alexandre's answer.

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  • $\begingroup$ "Middle ages" usually means up to the 15th century, which is not a time when merchants would "perform big calculations" with logarithms. A lot of the need for mathematical ease was for the demands of astronomy and navigation. In the late 1500s, prosthaphaeresis provided a method but it was largely abandoned once logarithms came into use. $\endgroup$ – Silverfish Jan 22 at 23:44
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Which came first, the natural logarithm or the base of the natural logarithm?

Quick answer: logarithms came before Euler's Number, $e$.

The Euler's Number, $e$, one of the most important mathematical constants is an irrational number closely related to growth and rate of change. The earliest written observation of number approximate to $e$ was made by J. Bernoulli, around 17-th century, arising from experimenting with the length and number of intervals of compound interest over an initial investment, where he observed a pattern which was later identified by Euler (and Gauss) as we know it Today.

The logarithms were developed, a century earlier (early 1600s) by Napier, as a practical tool for Astronomical calculations related with the multiplication of large numbers.

Around that time (middle 1600s) the concept of function became relevant together with Calculus, which is essentially the language of rate of change. Main part of that "language" is played by $e$ which arises naturally in expressions and functions related to growth. Calculus provided the "platform" that allowed $e$ to be associated and connected with another mathematical (already existing) branches - geometry (areas under a curve (hyperbole)), trigonometry, etc. leading to a culmination, which is named: "The most beautiful formula." (Euler's identity.): $$e^{i \pi} + 1 = 0$$ applicable and helpful in numerous areas in science.

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