Questions tagged [exponential-function]

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

530 questions
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The general solution of $x^a = a^x$ for real $a >0$

What are the roots of $$f(x) = x^a - a^x$$ for real $a > 0$? Case 1: For $0 < a < 1$ there is 1 solution, $x=a$. Case 2: For $1\le a < e$ there are 2 solutions: $x=a$ and $[x>a]$. Case ...
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If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute. The converse of this statement can be an easy application of the Cauchy ...
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How to prove that: $19.999<e^\pi-\pi<20$?

I would like to know how to prove $$e^\pi-\pi\sim 20.$$ More precisely, I want to show by using only mathematical tools that, $$19.999<e^\pi-\pi<20$$ I have checked with online ...
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Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to the ...
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Prove $e^x = \exp(x)$ starting with their limits-based definitions

Let $$e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n}$$ where $n \in \mathbb{R}$. Let $$\exp(x) = \lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^{n}$$ where where $n \in \mathbb{R}$ ...
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Why is $e$ the number that it is? [duplicate]

Why is $e$ the number that it is? Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference. ...
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Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
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Validity and Equivalence of two definitions of the real exponential function

The Problem : We state the following two definitions of the real exponential function from the Pr$\infty$fWiki page. We're interested in showing that the two definitions are valid $($i.e. the ...
Limit of $\frac{x^y-y^x}{x^x-y^y}$ while $x\to y$
$$\lim_{x\to y}\frac{x^y-y^x}{x^x-y^y}$$ I've tried L'Hospital and this. Both doesn't tends to work. Please help someone.
Limit involving $(\sin x) /x -\cos x$ and $(e^{2x}-1)/(2x)$, without l'Hôpital
Find: $$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$ I have factorized it in this manner in an attempt to use the formulae. I have tried to use that ...