Questions tagged [exponential-function]

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

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0
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1answer
16 views

Intersections between two columns of equal number points

How can I apply combinatorics to calculate the number of intersections between two columns of points? In the photo below let x be the number of point in each column, for x=2 we have 1 intersection, ...
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2answers
31 views

ln(e) + 4^(-2*(log(x)/(log(4))=(1/x)*log(100)) [on hold]

how would you solve Equation ln(e) + 4^(-2*(log(x)/(log(4))=(1/x)*log(100))
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1answer
51 views

Proof that the Fibonacci sequence grows exponentially

I wanted to write a proof similar to the one on page $3$ here ($2.2$ Fibonacci numbers): http://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf , to show that a sequence, defined ...
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1answer
40 views

Closed form of a function related to the exponential function

If $e^{-t}=\sum_{n=1}^\infty f(nt)$, is there a closed form for $f(x)$? And i dont mean anything like mobius inversion or $f(nt)=\frac {t^{n-1}}{\Gamma(n)}$ (This one doesnt even include ...
2
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1answer
27 views

Correct interpretation of exponential decay and decay factor

my question is about exponential decay and its factor. English isn't my native language and therefore I'm not sure about the precise definition in my particular case. I'm reading a specific paper ...
3
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3answers
89 views

Prove that $x^4e^x + y^4e^y \in \mathbb{Q}$ if…

If $e^x + e^y \in \mathbb{Q}$ and $xe^x + ye^y \in \mathbb{Q}$ and $x^2e^x + y^2e^y \in \mathbb{Q}$ and $x^3e^x + y^3e^y \in \mathbb{Q}$ Prove that $x^4e^x + y^4e^y \in \mathbb{Q}$ ...
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3answers
74 views

Why does each quadrant of the graph of $\sin(x)$ look like a section of an exponential function, variously rotated? [on hold]

The red section of $\sin(x)$ looks similar to the green $e^x$ (or $\frac{e^x}{200}-1$ to be precise). I'm aware that Napier based his logarithms on $\sin(x)$ in some sense, and I'm aware that the ...
2
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1answer
50 views

Equivalence of Two Forms of the Exponential Function [duplicate]

One encounters the following definitions for $e^x$ \begin{align} e^x &= \sum_{n = 0}^\infty \frac{x^n}{n!} \\ e^x &= \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^n \end{align} One can ...
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0answers
22 views

Calculate antilog digit by digit [duplicate]

Is there a formula/way how to calculate antilog digit by digit like this http://www.brics.dk/RS/04/17/BRICS-RS-04-17.pdf
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2answers
65 views

Solution of Diophantine Equations Using Calculus

Solve the equation in positive integers $$2^p+q^2=2^q+p^2$$ Here we see that $p=q$ is a solution otherwise let us assume that $q>p$, Now if we consider the function $$f(x)=2^x-x^2, x \geq 1$$ Now $$...
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1answer
38 views

Calculate logarithm and exponential using simple calculator [duplicate]

Is there a formula/way how to calculate logarithm precisely like log(3.587) or calculate 5^0.358 just using simple calculator.. I mean calculate precisely like scientific calculator not in estimation ...
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1answer
36 views

Is it possible to solve the two unknowns of this function given the area, the base length and a ratio between the start and end points?

I am trying to calculate the values of $a$ and $b$ in the following function: $$ f(x) = -e^{ax} + b + 1. $$ There are a few "rules" in play: $\int\limits_{0}^{n} f(x)\, \mathrm{d}x= v$ $f(n) = rb$ ...
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1answer
33 views

How can I solve $\int x\exp(-Tx^a)\,dx$?

How can I solve $$\int x\exp(-T x^a)\,dx$$ ($T$ and $a$ are variables.) In WolframAlpha, the answer is $$-x^2(Tx^a)^{-2/a}\,\frac{\Gamma(2/a,Tx^a)}a$$ I don't know why.
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1answer
52 views

Prove the following refinement of AM-GM inequality

Under what additional condition(s) the following inequality would be holds. In this case, prove your assertion and the inequality. For $x,y \ge 1$ (with $x \ne y$) we have \begin{align} \alpha x +...
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2answers
32 views

Simplify complex exponential

This is a solution from a textbook. I confirmed with an online calculator that this solution is true. $$ 1 - e^{-j*0.4\pi} = 1.176e^{j*0.3\pi}$$ How do you get from the left hand side to the right ...
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1answer
21 views

exponential decay of production

The production of a mine is decreasing exponentially,and in the past $5$ years there has been a decline of $18\%$.If production declines by $90\%$,the mine will close. The equation of production $P$ ...
2
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0answers
27 views

Solving this limit without employing l'Hôpital's rule to define the exponential derivative.

Reading this page to understand where does Euler's Theorem come from, the proof gets to a point where it tries to define the derivative of the exponent functions $f(x) = a^x$ as follows: First it ...
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1answer
11 views

How is compound growth affected by variations in interest?

When calculating compound interest often a constant interest rate is assumed. However, when applying this to dividend stocks for example, the dividend yield changes every year. This made me think ...
6
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1answer
81 views

Combinatorial proof that the exponential and logarithmic functions are inverse, the other way around

In the spirit of Combinatorial Argument for Exponential and Logarithmic Function Being Inverse, is there a combinatorial proof that $\exp$ and $\log$ are inverse, evaluating the composition the other ...
2
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1answer
87 views

I don't understand the logical leap made in the analogy of $e$

$e$ is often explained in terms of compound interest. If I found a bank that gave me 100% annual compound interest, then if I put in £1.00, at the end of the year, I would have £2.00. If I were more ...
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1answer
29 views

The function sequence $(\exp(f_n))$ converges uniformly if the function sequence $(f_n)$ converges uniformly.

I am trying to prove that if the function sequence $(f_n)$ on $X \subset \mathbb{C}$ converges uniformly to $f: X \rightarrow \mathbb{C}$, and the function $\textrm{Re}(f)$ (the real values of $f$) is ...
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0answers
12 views

Is there a way to simplify a n times elevated number with a different number as the last one?

let me explain what I mean since the title might be pretty confusing: $x^{y^{y^{.^{.^i}}}}$ There could be any number of y and I need to write this in a smaller way. Keep in mind that "i" must be ...
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1answer
30 views

Sum of exponential function inequality

For any positive a and n, it seems this inequality holds $$ \sum\limits_{t=n+1}^\infty e^{-at} \leq \frac{1}{a}e^{-an} $$ How can I prove this inequality and does this holds for negative a ?
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0answers
56 views

Help with expressing Partition Numbers as powers of two

Grateful for some help with expressing Partition Numbers as powers of two. I'm not great at maths, but this is as far as I have got: $P(n)=n/2+2^{n-1}-(2^{n-2})(2^x)$ where $x$ is an exponential ...
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1answer
38 views

Certain step in proof that sequence converging to $e$ monotonically increases.

I just can't get my head wrapped around a certain step in a proof that the sequence that approaches $e$ is monotonically increasing. The proof starts as follows: We want to show that: $x_{n+1} \geq ...
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1answer
48 views

Get around Exponential Integral in indefinite integral calculation?

I have two functions, $f(x) = (a_0*e^{(-a_6*(x + (a_1*x + a_2)))})$ and $g(x) = abs(a_3*x^2 + a_4*x + a_5)$ I'm trying to find the integral $\int{f(x)/g(x) dx}$. Wolfram gives back : $(a_0 exp(-1/2 ...
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2answers
59 views

Prove that the derivative of exponential is itself without using derivative [closed]

It's a little problem that I found interesting . To solve it without using differentiation we solve the following equation : Let $x,y$ be real numbers then solve : $$e^x=\frac{e^x-e^y}{x-y}$$ We ...
22
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2answers
545 views

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$?

Can $e^x$ be expressed as a linear combination of $(1 + \frac x n)^n$? In other words, does there exist an infinite sequence $(a_k)_{k \in \mathbb N_0}$ such that $$e^x = a_0 + \sum_{1 \leq k < \...
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1answer
56 views

Does $e^{xy}=(e^x)^y$ hold when $x$ and $y$ are real?

The question is as above. Does $e^{xy}=(e^x)^y$ hold when $x$ and $y$ are real? I remember that the answer is yes but am a little bit not confident. I know that the equality fails when $x$ an $y$ are ...
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2answers
109 views

Why can't the derivative of $e^{\ln x}$ be left as $\frac{e^{\ln x}}{x}$?

Yes, I know the answer is 1 because $e^{\ln x}$ is rewritten as $x^{\ln e}$, which is $x$, then multiplied by $\frac{1}{x}$ when taking the derivative of the inside function, resulting in the answer ...
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2answers
78 views

Solving $y'' + 2y' + 2y = 0$: How to eliminate imaginary unit from solution?

$$y'' + 2y' + 2y = 0$$ $\downarrow$ (write characteristic equation) $\lambda^2 +2\lambda + 2 = 0$ $\downarrow$ (solve characteristic equation) $\lambda = -1 \pm i$ $\downarrow$ (write general ...
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2answers
38 views

Euler's Formula explanation in $e^{ix} ⋅e^{iy}$ [closed]

According to Euler's Formula, $e^{ix} = \cos(x) + i\sin(x).$ I'm computing the product $e^{ix} \cdot e^{iy}.$ What is the real part (that is, the term without a factor of $i$)? Why is it $\cos(x)\...
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1answer
29 views

Finding inverse Fourier Transform

I have an expression like this: $$U(x,p)=e^{-ix(p^2/(2k))}\cdot e^{ix(k(m^2-1)/2)}$$ with $$U(x,p)=\int_{-\infty}^{\infty}u(x,z)e^{-ipz}dz$$ How do I find $u(x,z)$ I am not sure how to approach ...
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8answers
75 views

Prove that $\frac{e^{-a}}{1-e^{-a}}<\frac{1}{a}$ for positive $a$

The question is to prove the following inequality: $\frac{e^{-a}}{1-e^{-a}}<\frac{1}{a}$ for positive $a$. I have tried to make the RHS into the following form: $\frac{1}{1-(1-a)} = 1 + (1-a) + (...
2
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1answer
67 views

How to solve difficult exponential equation

I would like to know how can I solve the following exponential equation for $x$: $$\exp\left(\frac{n_1}{x}\right) + \exp(n_2) + \exp\left(n_3 - \frac{n_4}{x}\right) - \exp(n_5) = 0$$ where $n_1$, $n_2$...
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0answers
22 views

Variance of the exponents and sum of exponentials

Let's assume we have two sets of non-negative real-valued numbers $\{a_i\}_{i=1}^{N}$ and $\{b_i\}_{i=1}^{N}$ ($N$ can be large) such that $\frac{1}{N}\sum_{i=1}^{N}a_i=\frac{1}{N}\sum_{i=1}^{N}b_i=...
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3answers
46 views

How to find the inverse of $f(x)=x+(-1)^{x-1}$

Question: Find the inverse of $f(x)=x+(-1)^{x-1}$ where $f:N\to N$ ($N$ denotes set of natural numbers) Usually, when I am asked to find inverse of a given function, I use to express the ...
3
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2answers
89 views

Prove that $\exp : \mathbb{C} \to \mathbb{C}$ is periodic using as little geometry as possible.

The proof attempt below is long, please consider it to be an attachment of sorts. The main content of the question is the introduction paragraph, and the proof attempt is my attempt to answer my ...
1
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3answers
33 views

Question about holomorphic functions and zero's in complex analysis

The question: Write an example of a connected, open set $U ⊂ C$ and an analytic function $f : U → C$ such that $f(z)$ has $N$ zeros on $U$ while $f′(z)$ never vanishes on $U$ My guess is that I was ...
0
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1answer
33 views

Solve for t in overdamped 2nd order DE with non-integer constants

How would I rearrange an equation like this where there are 4 unique constants that are non-integers in front of the exponent and in front of e so as to solve for t? $$x(t) = Ae^{Bt} + Ce^{Dt}$$ If ...
0
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0answers
32 views

Integral resulting to an incomplete gamma function

I want to evaluate the following integral $$I(x)=\int_{-1}^1 \bar{x}^n\,\mathrm{e}^{-\lambda\,|x-\bar{x}|}\,\mathrm{d}\bar{x}$$ with $x\in\mathbb{R}$, $n\geq0$ and $\lambda\in\mathbb{C}$. First ...
0
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1answer
41 views

Differential equation for bacterial growth

I am assigned with a question which states the rate of a microbial growth is exponential at a rate of (15/100) per hour. where y(0)=500, how many will there be in 15 hours? I know this question is ...
1
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1answer
43 views

Limit of the quotient of exponentials [closed]

Let $x_n,y_n$ be two sequence with $$\lim_n\frac{x_n}{y_n}=1.$$ It is true that $$\lim_n\frac{e^{x_n}}{e^{y_n}}=1?$$
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2answers
46 views

Solving exponential simultaneous equations with 4 unknown coefficients. [closed]

While solving for a best-fit function with the general form of an exponential equation, that is, $$y=ab^{x-c}+d,$$ I reached these following simultaneous equations as below: \begin{align} ab^{9-c}+d &...
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0answers
32 views

Inequality involving double summation

I have the inequality $$e^{-10}\sum_{a=0}^{n-i}a\sum_{b=0}^{n-1}e^{-abn(a-2bn)}<\ln n$$ A table in Mathematica quickly shows that the largest integer $n$ for which the inequality holds is $n=2$. ...
0
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1answer
41 views

Solve inequality involving double summation, exponentiation and $\ln$

I have the following inequality (where $n$ is a real number): $$\sum_{a=1}^n\sum_{b=1}^n\frac{a^n}{n^2}e^{-n-ba}\ge\ln n$$ Computation suggests this holds for $n$ greater than or equal to a number ...
0
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1answer
43 views

How to work exponential function backwards from result?

There is a question that is asking me T=Temperature(Celsius) (t=minutes since poured in) where the function for finding temperature of coffee is $$ T=90(3^{-0.05t})$$ (initial coffee temp is 90) it ...
-1
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1answer
60 views

Prove that $n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$ [closed]

Prove that if $s_n=1^1+2^2+3^3+\cdots+n^n$ then $$n^n \left( 1 + \frac{1}{4(n-1)} \right) \leq s_n < n^n \left( 1 + \frac{2}{e(n-1)} \right)$$ (it holds for $n$ larger than $2$). I want to prove ...
8
votes
2answers
142 views

Asymptotic behaviour of $f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!}$ for $\varepsilon\in(0,1)$

Let $\varepsilon \in (0, 1)$ and consider the analytic function $$f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!}.$$ What is the order of growth of $f(x)$ as $x \to \infty$? From the basic ...
2
votes
2answers
44 views

Are there functions $f$ and $g$ such that $f \circ g$ is exponential even though neither $f$ nor $g$ is?

To be precise, call a function $f$ exponential if $f(n) = \Omega\bigl(2^{n^c}\bigr)$ for some constant $c > 0$. Some motivation: composing two exponential functions yields a doubly exponential ...