Questions tagged [exponential-function]

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

The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance).

Definition: If $~b~$ be any number such that $~b\gt 0~$ and $~b\neq 1~$ then an exponential function is a function in the form,$$f(x)=a~b^x$$ where $~b~$ is called the base , the exponent,$~x~$ can be any real number and $~a\neq0~$.

${}$

Properties:

  • The graph of $~f(x)~$ will always contain the point $~(0,1)~$. Or put another way, $~f(0)=1~$ regardless of the value of $~b~$.
  • For every possible $~b~$we have $~b^x\gt 0~$. Note that this implies that $~b^x\neq 0~$.
  • If $~0\lt b\lt 1~$then the graph of $~b^x~$ will decrease as we move from left to right. Check out the graph of $~\left(\frac{1}{2}\right)^x~$ above for verification of this property.
  • If $~b\gt 1~$ then the graph of $~b^x~$ will increase as we move from left to right. Check out the graph of $~2^x~$ above for verification of this property.
  • If $~b^x=b^y~$, then $~x=y~$.

${}$ The Natural Exponential Function: In mathematics, the natural exponential function is $$f(x)=e^x~,$$ where $e$ is Euler's number.

Note: $f(x)=e^x~$ is a special exponential function. In fact this is so special that for many people this is THE exponential function.

Applications:

Exponential functions are solutions to the simplest types of dynamical systems. It is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative.

References:

https://en.wikipedia.org/wiki/Exponential_function

http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U18_L1_T1_text_final.html

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Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
Jichao
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Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
Superbus
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How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet $$\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$$ which looks like unbelievable. (I forgot…
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Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}\tag1$$ Now, $$f'(x)=\frac{e\pi(1-\ln…
mathlove
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What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. The question is motivated from the simple exercise…
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From $e^n$ to $e^x$

Solve for $f: \mathbb{R}\to\mathbb{R}\ \ \ $ s.t. $$f(n)=e^n \ \ \forall n\in\mathbb{N}$$ $$f^{(y)}(x)>0 \ \forall y\in\mathbb{N^*} \ \forall x\in\mathbb R$$ Could you please prove that there exists an unique solution: $f(x)=e^x$? (Anyway,…
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Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later edit: . . . where $e$ is defined as the base of an exponential function…
Michael Hardy
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Intuitive explanation for why $\left(1-\frac1n\right)^n \to \frac1e$

I am aware that $e$, the base of natural logarithms, can be defined as: $$e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$ Recently, I found out that $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n = e^{-1}$$ How does that work? Surely the minus…
Bluefire
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Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of it?
Ray Kay
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New bound for Am-Gm of 2 variables

Today I'm interested by the following problem : Let $x,y>0$ then we have : $$x+y-\sqrt{xy}\leq\exp\Big(\frac{x\ln(x)+y\ln(y)}{x+y}\Big)$$ The equality case comes when $x=y$ My proof uses derivative because for $x\geq y $ the function…
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Difference between power law distribution and exponential decay

This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like the one in the Power law article, which is the…
user19821
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Prove that $(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$

Prove that $f(x)=(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$ for all $x>0.$ We have $f(x)=f(\frac{1}{x}), f'(x)=-\frac{1}{x^2}f'(\frac{1}{x}),$ so we only need to prove $f'(x)>0$ for $0 < x < 1.$
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How do I interpret Euler's formula?

I don't understand the formula at all: $$e^{ix} = \cos(x) + i \sin(x)$$ I've tried reading all sorts of webpages and answers on the subject but it's just not clicking with me. I don't understand how we can define things when these are already known…
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Proof of the derivative of $\ln(x)$

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) - \ln(x)}{h} \\ &= \lim_{h\to0} \frac{\ln(\frac{x +…
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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 know how to start with. (this problem was shared in…
user153330
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