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Feel free to close this question if it's a duplicate, but my search did not yield the desired result. Let's say I only know that $\mathrm e = \lim_{x\to 0}(1 + x)^\frac1x$. How can I show that $$ \lim_{x\to 0}\mathrm e^{\frac{a^x - 1}{x}} = a? $$ That must be very basic, but I'm missing it.

SBF
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    So how do you define $\log$ please? I mean do we know that for $a>0$, $e^{\log(a)} = a$ or not ? – Axel Oct 23 '22 at 09:44
  • @Axel yes; we can rephrase even as $\lim_{x\to 0} \mathrm e^{\frac{a^x - 1}{x}} = a$ – SBF Oct 23 '22 at 09:49
  • Ok, and I guess we know nothing of the differentiabilty of $\exp$ then? – Axel Oct 23 '22 at 10:05
  • Do you mean the "inverse" of the exponential function, not derivative? – obareey Oct 23 '22 at 10:05
  • @Ilya Axel is asking which definition of $\ln x$ you are taking. There are multiple non-equivalent definitions, including $\ln x = \int_1^x \frac{dt}{t}$ or $\ln x =(x-1)-(x-1)^2/2+\ldots$ or $y=\ln x\iff x=e^y$, etc. Until you specify which definition you are using, your question is unanswerable. Defining $e$ alone does not define $\ln x$. – Jam Oct 23 '22 at 10:07
  • @Jam sure, I gave an affirmative reply to Axel's question whether we can use that $\log$ is the inverse of $\exp$ – SBF Oct 23 '22 at 12:57
  • @Axel we know nothing, yes. Neither whether derivative exists at any given point, nor its value. – SBF Oct 23 '22 at 12:59

2 Answers2

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The basic idea is not that different from the usual calculation of the function inverse. Let's say you have a function $y = f(x)$ and you want to calculate $x = f^{-1}(y)$. In your case the function should be (see also https://math.stackexchange.com/a/2432906/111671)

$$a = f(u) = e^u = \lim_{x \to 0} (1 + x)^{u/x} = \lim_{x \to 0} (1 + ux)^{1/x}$$

Now, you want to write $u$ in terms of $a$ to get $u = f^{-1}(a)$. This is the main idea but a rigorous proof may be a bit more involved since it contains limits.

obareey
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This is a good question.

First, you need to define the function $e^x$. But by considering first integer and then rational values of $x$, you can define for any real $x$, $$ e^x = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n $$ and check that it obeys the expected properties of exponents. Then, using these properties, you show that $$ \frac{d}{dx}e^x = e^x $$ and therefore $e^x$ is strictly increasing. This allows you to define $\ln x$ as the inverse function. Now, you can define $$ a^x = e^{x\ln a} $$ The chain rule now gives you what you want.

Deane
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