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A friend and I were discussing the "early transcendentals" terminology used on some calculus books and looked up the phrase, which directed me to this answer:

What's the difference between early transcendentals and late transcendentals?

It resparked some questions I've had about logarithmic and exponential functions for a long time, mainly about their history. My understanding is that logarithms were conceived and utilized for calculation long before Euler's constant was used, and that in their early years, the connection that logarithms are inverses of exponentials was not made. Am I understanding that correctly? If so, it's always puzzled me. If not defined in the context of exponentials, how did logarithms arise historically, and how were they originally defined? What were they used for?

Similarly, what did the historical development of $e$ look like? Because there are so many equivalent ways of talking about it, its not clear to me how it may have originally come about or why, and what it's original definition looked like. I've seen it defined as $\lim_{n\to\infty}(1+\frac{1}{n})^n$, I've seen it defined as the base of the natural logarithm (but how would the natural logarithm have been defined without $e$?) I've seen it accepted as the value of $\sum_{n=0}^\infty\frac{1}{n!}$, etc. I know these definitions are all equivalent, but what was the originally proposed definition for $e$ and what drove the development of its theory (and consequently all the equivalent definitions)?

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  • $\begingroup$ You might look at this MacTutor page $\endgroup$ – Robert Israel Apr 29 '17 at 2:47
  • $\begingroup$ I think everything started with the "area under an hyperbola" problem. They found that $\int_1^{ab} \frac{dt}{t} = \int_1^{b} \frac{dt}{t}+\int_1^{a} \frac{dt}{t}$ by some geometric methods, before they even could define integrals. So this area $\int_1^a \frac{dt}{t} $ was a logarithm, in base "$e$". $\endgroup$ – reuns Apr 29 '17 at 3:43
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    $\begingroup$ A few years ago, I wrote an answer for logarithms. But an even better account, that gets to the heart of the matter for the motivation for logarithms, is here. ($e$ appeared there, but the connections to the rest of calculus came much letter.) $\endgroup$ – ShreevatsaR May 5 '17 at 9:39
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I'm not sure this question belongs here. But $e$ was first found when taking the derivative of exponential functions. they found $$\frac{d}{dx}a^x=ln(a)a^x$$ and decided that when $ln(a)=1, a=e$, this means that $\frac{d}{dx}e^x=e^x$ is what defined $e$ originally

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