# Tagged Questions

For question involving exponential functions and questions on exponential growth or decay.

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### What's so “natural” about the base of natural logarithms?

There are so many available bases. Why is the strange number $e$ preferred over all else? Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
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### Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
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### Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
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### Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
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### Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
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### If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Addionally, what ...
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### Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
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### Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't even ...
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### Is there a (deep) relationship between these various applications of the exponential function?

Here is a list of some applications of the exponential function. 1) The exponential mapping in Lie theory. I put this first because my intuition tells me that this must be the most fundamental, or ...
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### $\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
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### How to show $\frac{19}{7}<e$
How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...