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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Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because…
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The math behind Warren Buffett's famous rule – never lose money

This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first. Warren Buffett famously provided 2 rules to investing: Rule No. 1: Never lose money. Rule No. 2: Never…
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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 to any student what kind ideas already used but…
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Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
Billy Thompson
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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 canonical proof? I would ideally like a proof which…
Ashley Montanaro
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Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
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How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to prove this theorem.
faceclean
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If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

Let $a$ and $b$ be positive numbers such that $a+b=1$. Prove that: $$a^{4b^2}+b^{4a^2}\leq1$$ I think this inequality is very interesting because the equality "occurs" for $a=b=\frac{1}{2}$ and also for $a\rightarrow0$ and $b\rightarrow1$. I tried…
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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?
user218
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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 positive irrational numbers $a\ne e$ such that…
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Why does an exponential function eventually get bigger than a quadratic

I have seen the answer to this question and this one. My $7$th grade son has this question on his homework: How do you know an exponential expression will eventually be larger than any quadratic expression? I can explain to him for any particular…
John L
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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 happens in $M_n(\mathbb{R})$? Nota Bene: As a…
Seirios
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Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=1$$ $$\lim_{n \to \infty}…
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Intuitive Understanding of the constant "$e$"

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you…
sova
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About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
Mai09el
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