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Natural exponential function is defined as an exponential function with base $e$. Then natural logarithm is defined as an inverse of that.

But then natural logarithm is defined as an integral.Then natural exponential function is defined as an inverse of that.

How is it beneficial?

Vedant Chhapariya
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    You have to start somewhere. Defining $\ln(x)$ immediately defines a function. After that, you just have to prove that it has the right properties. Another approach is to define $e^x$ as a power series. – lulu Nov 13 '21 at 12:30
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    Because we have less preliminary work to do - the other definition means we first have to define $e^x$ and then show that that function has an inverse... – David C. Ullrich Nov 13 '21 at 12:31
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    To stress, referring to "an exponential function with base $e$" requires you to define $e$ (not so easy) and then you have to prove that $e^x$ exists (also not so easy). – lulu Nov 13 '21 at 12:31
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    In fact, the exponential function $e^x$ also has more than one definitions, the limit one, the series one, by a differential equation, and one uses whichever is 'beneficial' in that context. All you need to see is whether or not your definition can be used to show the other definitions. – ultralegend5385 Nov 13 '21 at 12:38
  • No I mean to say that the first one seems to more natural as e is just a base of exponential function whose derivative is equal to itself. Then $ln x$ is just it's inverse. But in second, how would someone even come up with definition of $ln x$ as an integral? – Vedant Chhapariya Nov 13 '21 at 12:38
  • A related question: https://math.stackexchange.com/questions/2678805/how-can-we-come-up-with-the-definition-of-natural-logarithm – awkward Nov 13 '21 at 13:44

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