Questions tagged [exponential-function]
For question involving exponential functions and questions on exponential growth or decay.
8,059 questions
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One exponent for all numbers [closed]
We can say for given $a,b,c$ value,
$a^b=a^c$
Since base is equal, we can conclude
$b=c$
,but why this not valid for $a=1$,what is the intuitive idea?
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The closure of the set of triples $(x,y,z)$ admitting non-negative integer exponents with $x^{n_1}+y^{n_2}=z^{n_3}$
I am considering the set $S'$, which extends the original problem to allow non-negative integer exponents:
$$
S'=\{(x,y,z)\in \mathbb{R}^3:\exists n_1,n_2,n_3\in\mathbb{N}_{\ge 0},\ x^{n_1}+y^{n_2}=z^{...
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Exponential function of negative argument [closed]
Famous $e^x$ function obeys the following well-known property:
$$
e^{-x} = \frac{1}{e^x}.
$$
I am concerned about the following. What if we didn't know that property in advance. Is it possible to ...
- 551
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Is there any closed form for the solution of the equation $\exp(x) - x = 2$ [closed]
I randomly gave this as a problem to myself and it seems to not be solvable by basic equation manipulation. Is this possible to solve?
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Explain why the function $\exp(z^2)$ has an antiderivative on the whole complex plane. [closed]
I'm working through an undergraduate course in complex analysis and am currently on a chapter entitled Cauchys integral theorem, where i encountered this question as an exercise. It doesn't seem like ...
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Find $ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$ [closed]
I see the proccedure of:
How I find the limit of $\frac{2^n}{e^{p(n)l}}$
I didn’t understand how it applies in my case:
$$ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$$
$$ \lim_{x\to\infty} \frac{\...
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Found a strange differential equation with quaternions and $\exp()$, how do I solve?
Found this problem on a blackboard today. It looks like some form of quaternion analysis, something to do with the MacLaurin series of $e^x$ My question is, what all is going on here, and how might I ...
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4
answers
956
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Prove lower bound of complicated function to show divergence at 0
I have the function
$$g(\theta) = \frac{1}{2 \pi} \int_0^{\infty} \frac{1}{c}\text{exp}\left(-\frac{\theta^2}{4}c\right) \text{exp}\left( -\frac{1}{c}\right) dc$$
and I want to prove that as $|\theta| ...
- 219
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Rigorous proof that monodromy group of logarithm is $\mathbb{Z}$
I have seen a lot of references on the topic but none of them really contained a proof from first principles that the monodromy group of logarithm is $\mathbb{Z}$. Here's the developed theory in my ...
- 2,989
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4
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Limit of $ e^{x(\ln2 - x)}$
Recently I came across the limit
$$
\lim_{x\to\infty}\frac{2^x}{e^{x^2}}
$$
I have organized it to be
$$
\lim_{x\to\infty}\frac{e^{x\ln2}}{e^{x^2}}
$$
Now, I want to say that by comparing the ...
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Relating $\sum_{k=-\infty}^{\infty}e^{-(x-2k)^2}$ and $\cos(x)$
I was playing around with function of the type $f(x)= e^{-(x-k)^2}$.
Then I noticed that $G(x)=\sum_{k=-\infty}^{\infty}e^{-(x-2k)^2}$ had the osccilating nature of a cosine wave, which was streched ...
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Find maximum of $3^ba^{b+1}$ for positives $a+3^b=6$ using derivatives
The following problem is from a contest in Yunnan, China.
If positive real numbers $a$ and $b$ satisfy $a+3^b=6$, find the maximum of $3^ba^{b+1}$.
Here is one solution: let $a=3^c$, then by AM-GM,
$...
- 4,882
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naming family of functions defined by average of exponential functions w.r.t. rates
I have a family of functions defined by "a weighted average of exponential functions across rates from any probability distribution", or mathematically:
$$ f(t) = \int e^{-t\lambda} p(\...
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Taylor expansion at an apparently discontinous point
For $t\neq 0$, we define $g(t)$ as follows
\begin{align}
g(t)=\frac{1}{t}
\end{align}
so we have
\begin{align}
g'(t) = -\frac{1}{t^2}
\end{align}
Then we define $G(t)$ with
\begin{align}
G(t)&=g'(...
- 191
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Can I cancel or negate exponential bases with the same exponent (e.g. in Euler’s relation or other contexts)?
I recently came across Euler’s relation on the inside cover of my "Classical Mechanics" physics textbook by Taylor:
$$
e^{ix} = \cos(x) + i\sin(x)
$$
That made me curious about how ...
2
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5
answers
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Prove $h(t)=(1-t)^{\frac{n-1}{2}}-\mathrm{e}^{-\frac{n}{2}t}\geq0$ for any $t\in[0,\frac{1}{n}]$
The question is to prove $h(t)=(1-t)^{\frac{n-1}{2}}-\mathrm{e}^{-\frac{n}{2}t}\geq0$ for any $t\in[0,\frac{1}{n}]$, where $t\in\mathbb{R}$ and $n$ is a positive integer. I list some of my findings ...
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Prove that $\ln x\le x^{x^x}-x^{1-x}$
Using $e^x\ge x+1$, prove that $$\ln x\le x^{x^x}-x^{1-x}$$
Since $\ln x\le x-1$, it is enough to prove $$x-1\le x^{x^x}-x^{1-x}$$
or
$$x^{x^x}-x^{1-x}-x\ge-1$$
or
$$x+x^{1-x}-x^{x^x}\le 1$$
Maybe, ...
2
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3
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Help with a year 12 algebra problem for investigation: $10 \, t \, e^{-t} + 10 \, (t-6) \, e^{6-t} = 1.1$
I have tried to use symbolab to solve this problem $10 \, t \, e^{-t} + 10 \, (t-6) \, e^{6-t} = 1.1$, and it does something with limits and, as i do not have the pro version, I cannot access the step ...
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What is the unit's digit of $a^b,$ where $a$ is a rational number and $b$ is an integer?
Given $a^b$, where $a$ is a rational number and $b$ is an integer, what is the value of the unit's digit?
Question Origin
After showing someone how to find the unit's digit of $a^b$ given $a$ and $b$ ...
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How can we prove the inequality $x - x^{1-x} + x^{-x^{x-1}} \ge 1$ using Bernoulli's inequality?
For all $x>0$, prove that: $$x - x^{1-x} + x^{-x^{x-1}} \ge 1$$
We must proceed using only Bernoulli's inequality and some elementary inequalities.
I tried to manipulate the inequality a bit by ...
2
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1
answer
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Find $\min\limits_{a_{25}<b_t}\{t\}$ where $\{a_n\}$ and $\{b_n\}$ are exponentially recurrent.
Let the sequence $\{a_n\}$ be such that $a_1=2$, $a_{n+1}=2^{a_n}$, and the sequence $\{b_n\}$ be such that $b_1=5$, $b_{n+1}=5^{b_n}$. Find the smallest subscript $t$ such that $a_{25}<b_t$.
I ...
- 4,882
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Prove that $e^x > x^x y^y (y + 1)$, where $x$ and $y$ are positive real numbers satisfying $x^x + y^y + y - x = 2$
Problem$\,$ Let $x$ and $y$ be positive real numbers such that $x^x+y^y+y-x=2.$ Prove that: $$e^x>x^xy^y(y+1)$$
This problem is about the application of the popular inequality $e^x\ge x+1.$ So, we ...
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1
answer
162
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Continued fraction for $e^x$
I recently started studying continuous fractions and came across some decompositions whose origin is unclear to me. For example, for $e^x$:
$$e^{x}=1+\frac{2 x}{2 -x+\dfrac{x^2}{6 +\dfrac{x^2}{10 +\...
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Prove that $\left(1+\frac ab\right)\left(1+\frac ab+\frac bc\right)\left(1+\frac ab+\frac bc+\frac ca\right)\ge16$ [duplicate]
Let $a$, $b$, $c$ be positive real numbers. Prove that
$$\left(1+\frac ab\right)\left(1+\frac ab+\frac bc\right)\left(1+\frac ab+\frac bc+\frac ca\right)\ge16.$$
This was a contest problem from ...
- 4,882
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Sum of exponential of monotonic sequence.
Consider a sum of the form
$$\sum_{n=0}^N e^{a_n} \quad , \tag{1}$$
where $a_n$ is some monotonic sequence converging to zero. I am assume that because of its generality, there are little to none ...
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