# Questions tagged [exponential-function]

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

5,196 questions
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### Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
2answers
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### Unintuitive behavior of x^y for x<0

I take advantage of mathematics a lot while programming, but in spite of that I'm very mathematically ignorant. Sorry if I'm asking something stupid. Often I need to exponentially amplify some values....
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### The matrix exponential [on hold]

Prove that $$\left(e^A\right)^{∗}=e^{A^∗} .$$ I've tried messing around with both sides, I just can't get the two to match up. Any ideas?
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### Why is the solution to dy/dx = y exponential?

I am trying to understand how e was originally derived as a solution to the below differential: I understand how we reach: For my sake let's call the above solution to the differential f(x) instead ...
1answer
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### For $a<b<c$, is it true that $c^n>a^n+b^n$ for large enough $n$?

Given three numbers $a$, $b$, $c$ with $a < b < c$, does the following hold? $$c^n > a^n+b^n \qquad \text{ for large enough }n$$
0answers
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### Solving exponential and logarithm mixed together

Exp(-x)=Cosx. So i shifted the exponent to the right hand side. 0=exp (x)Cosx. Got stuck here. Don't know whether to solve them individually. Like Cosx =0 Exp (x)=0
8answers
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### Show $\sin(\frac{\pi}{3})=\frac{1}{2}\sqrt{3}$

I have to show that $$\sin\left(\frac{\pi}{3}\right)=\frac{1}{2}\sqrt{3}$$ and $$\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}$$ Should I use the exponential function?
25answers
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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 ...
1answer
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### Definite integral involving modified bessel function of first kind, exponential of higher order [on hold]

Is there a solution of integral of the following form: $$\int_0^\infty e^{-x^n} J_0(x) dx$$ I have tried searching a lot but cannot find anything. Any kind of help will be appreciated.
1answer
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### Is exponential growth and decay faster than polynomial growth and decay?

I know the answer is yes for growth conditions, but I don't see how it's obvious that exponential decay is faster than polynomial decay, say for a polynomial $x^2$.
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