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

learn more… | top users | synonyms (1)

0
votes
0answers
27 views

Measuring the spread of epidemics

Imagine you were tasked with spreading an epidemic (or the cure to one, if it makes it easier, same math). This spread happens in one-on-one meetings happening every ten minutes. Each person in the ...
2
votes
1answer
33 views

What is the name of this function, $f(x) = \frac{1}{\exp(-kx)+1}$?

What is this function, $f(x) = \frac{1}{\exp(-kx)+1}$, where $k$ is a constant, called?
0
votes
0answers
118 views

Discretization of continuous system dynamics

Assume a system $$ \dot x = A x + Bu, \qquad x\in\mathbb R^n, u\in\mathbb R^m. $$ Now I want to calculate the matrices $A_d$ and $B_d$ such that the discrete system with sampling interval $T$ $$ ...
0
votes
1answer
40 views

Intervals of Convex and Concave function

Find the intervals where the function is convex and concave. $$f (x) = e^{2x} - 2e^x$$ I tried differentiating twice, and my answer is: concave when $x < \ln (1/2)$ and convex when $x > ...
0
votes
1answer
39 views

Is it possible to estimate $e$ based on $N$?

Consider a sequence of random numbers $u_1,\dots,u_n$ obtained from a continuous distribution $F$. Let $N$ be the first one that is greater than its immediate predecessor. In othe words, ...
1
vote
2answers
67 views

What is the inverse of $f(x) = a⋅e^{bx} + cx + d$

Does an inverse function for $f(x) = a⋅e^{bx} + cx + d$ exist where a, b, c, d are constants? If so, what is it? I've tried lots of methods, but they've all failed. What I ended up doing to ...
0
votes
1answer
34 views

Use given identity to computer exponent of 4x4 matrix

I've been given an identity (that I don't know how to prove unfortunately), and been asked to use it to compute exp$(xM)$, where $$ M = \begin{bmatrix} 1 & 1 & 1 & 1 \\ ...
0
votes
0answers
28 views

Why does this derivation of exponential growth give a different, but apparently not wrong, answer?

Here's a fairly standard derivation of the exponential growth equation. $\frac{\text{d}x}{\text{d}t} = kx$ $\int\frac{1}{x}\text{d}x = \int k\text{d}t$ $ln(x)=kt+C$ $x=C'\text{e}^{kt}$ Right? ...
0
votes
0answers
61 views

Proof of a generator for coprime integers

Take the integers coprime to $p$ (all but multiples of $p$). Does there always exist an integer (generator) $a$ coprime to $p$ that generates the entire group of coprime integers under powers of $a$? ...
1
vote
2answers
45 views

Integer solutions to an exponential equation

Are there any integer solutions for the equation $$2^x+2=5^y$$ Similarly, are there any solutions to $$2^x-2=5^y$$ I ask the second because I'm not sure if they are answered similarly. Put ...
1
vote
1answer
52 views

Keeping an exponentially decaying system steady.

To give a bit of background: I am trying to figure out what amount of substance X to continuously add over a time interval in order to keep it constant in a system where substance X has the half-life ...
1
vote
0answers
36 views

Does Euler's recurrence relation for partitions imply that the partition function grows exponentially

Can one, just by manipulating the series, demonstrate that the partition function must be growing exponentially or at least that it is unbounded by any polynomial? If so, then how would it be done. ...
0
votes
0answers
19 views

General equation for specific rotated and translated exponential function through two points

Given the exponential function yα(x) = AeB(x-x0)+y0 that passes through points P0=(x0,y0) and P1=(x1,y1), I'm trying to find a function yβ(x), which passes through the same two points and ...
2
votes
1answer
38 views

For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
1
vote
1answer
30 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
1
vote
1answer
27 views

Calculating limits using the definition of number e

I have some examples in Demidovič using this technique and there seems to be no reliable source for them online, so I'll make a small tutorial. Example 1: ...
0
votes
2answers
56 views

What equation has the form f(x) = n exp(m x)?

I'm a programmer working on a calculation with a curve trend. I'm using OpenOffice Calc (like MS Excel) and it's given me a formula for a graph that I don't understand. I can't find this form ...
8
votes
6answers
181 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
3
votes
1answer
78 views

Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
1
vote
1answer
50 views

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $ with $ q \geq 2 $ I'm not sure how to approach this question. I was thinking through in induction with ...
1
vote
2answers
37 views

Is the exponential function continuous for complex numbers?

Hey this might be a dumb question so here it goes: Is $e^{(x)}$ continuous for $x\in \mathbb{C}$? Specifically this question arose while solving the differential equation in the form of ...
2
votes
1answer
59 views

Find all real solutions for $x$ in $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 .$

Find all real solutions for $x$ in $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 .$ I have found out that the answers were 0,1, and -1. But I used sort of a guess-and check way. ...
0
votes
0answers
18 views

How to steepen logarithmic function without reducing constant of deceleration

As you can see I have plotted my points in Geogebra and compared them to the function $ y=log_{10}x $ They clearly don't coincide, how would I go about adjusting the function in order to find the ...
0
votes
1answer
37 views

Integration of $\int_{-2}^{\infty} k^m e^{-a k^2} dk$

How to solve the definite integration as showed in the title. And $m$ is an arbitrary natural number, $a$ is a non-negative number. Many thanks in advance.
3
votes
4answers
360 views

Exponential Growth

I'm trying to wrap my head around the algebra used to get a solution. The question states: In 2011, the Population of China and India were approximately 1.34 and 1.19 billion people, ...
2
votes
2answers
51 views

Solution of equation $[1+\frac{x}{b}]e^{-x/b}=z$

Can we solve this equation $$\left(1+\frac{x}{b}\right)e^{-x/b}=z$$ We have to determine value of $x$ in term of $z$. Problem occur while calculating the following integral. ...
1
vote
0answers
21 views

Exponential distribution help

Terry has a part-time job at a call centre. Calls to the call centre occur according to a Poisson process with rate $\lambda$ calls per minute. Terry decide to measure the time that elapses between ...
0
votes
1answer
33 views

How to solve equations containing logarithms and exponentials

Equation 1: $x+e=e^x$ According to Wolfram alpha : Solution of x $\approx$ -2.6 and 1.4 Equation 2: $x-e = \ln(x)$ According to wolfram alpha, Solution for x $\approx$ 0.07 and 4.1 How does ...
0
votes
2answers
114 views

Prove that factorial grows faster than exponential function using limits [duplicate]

How can I prove that the factorial function ($n!$) grows faster than exponential functions (ex: $2^n$) using limits?
0
votes
1answer
55 views

Determine value $b$ in $f(x)=ab^x$ given the following data points

If $f(x)=ab^x$, what is the value of $b$ if $(0,35)$ and $(3,125)$ are data points? Is this the way to do it? $$35=ab^0,$$ $$a=35.$$ $$125=ab^3,$$ $$125=3\log(35)+\log(b),$$ ...
0
votes
0answers
38 views

Best goodness-of-fit measure to determine: is my dataset power law or exponentially distributed?

I would like to determine whether a discrete dataset that I'm modeling would be better fitted by an exponential function or a power law function. I'm aware that a chi-squared test may be a suitable ...
1
vote
1answer
35 views

Simplifying the Pauli matrix expression $e^{-i\sigma_x\phi/2}$

As in the title, the expression is: $$e^{-i \sigma_x \phi/2}$$ Where $\sigma_x$ is: $$\left\{ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right\}$$ Where would I even begin in simplifying ...
1
vote
1answer
51 views

$\exp(i \theta)=1?$

So I was thinking, $\exp(i\theta) = \exp( i\theta\cdot2\pi\cdot\frac{1}{2\pi})$, we can rearrange it, so that: \begin{align} & \exp\left( i\theta\cdot2\pi\cdot\frac{1}{2\pi}\right)=\exp\left(2\pi ...
1
vote
3answers
86 views

How to prove this? $ \lim_{x \to 0}\frac{e^x-1}{x}=1 $ [duplicate]

Any idea how do I prove the following? $$ \lim_{x \to 0}\frac{e^x-1}{x}=1 $$ Thanks
1
vote
4answers
92 views

How to determine the monthly interest rate from an annual interest rate

I have a calculation which gives me the annual interest rate if I already know the monthly interest rate as follows: (Monthly interest rate + 1)^12 In this case I ...
2
votes
0answers
34 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
1
vote
1answer
49 views

Solving natural logarithms with absolute value

Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
0
votes
1answer
28 views

Simplification involving exponents to base e

I've found the following expression. It looks really simple - so it's driving me crazy, that I don' get it: $(e^{3x}).(2)$ is simplified as $2e^{2x}$. Similarly, $(2x+7).(3e^{3x})$ is simplified as ...
1
vote
4answers
53 views

Prove $f'(z)=f(z)$ implies $f(z)=ce^z$ for some $c \in \mathbb{C}$ [duplicate]

Suppose that: $$f'(z)=f(z) \text{ for all }z\in\mathbb C.$$ In other words, the complex function $f$ is equal to its own derivative. Prove that there is a constant $c\in\mathbb C$ such that $f(z)=c ...
6
votes
0answers
59 views

Exponential fields as structures with three binary operations.

The exponential rings and fields are usually studied as structures with two binary operations $(+,\cdot)$ and one unary operation $\exp(x)$ defined on a set $K$. Why not consider the exponential as a ...
2
votes
2answers
81 views

Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime ...
1
vote
1answer
43 views

What is the remainder of $|e-\sum_{j=0}^n{1\over j!}|$?

I have to find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$, but I want to do it with the remainder. I know that it is ${e^c\over (n+1)!}$ where $0<c<1$, but how do you get ...
2
votes
1answer
91 views

System of exponential equations

If $x,y,z \in \mathbb{R}$ and $$ \begin{cases} 2^x+3^y=5^z \\ 2^y+3^z=5^x \\ 2^z+3^x=5^y \end{cases} $$ does it imply that $x=y=z=1$?
0
votes
1answer
30 views

Half life, exponential decaying equation question

If a radioactive substance has a half-life of $10$ days, in how many days will $1/8$ of the initial amount be present? Assume the decaying process is continuous (exponential). Will the answer just be ...
5
votes
4answers
901 views

Euler's formula, is this true? [duplicate]

*I've changed this question as below. Let me have a function such as $ f(k) = \exp(j 2 \pi k ) $, where $k$ is real value. Using Euler's formula, we can write $f(k)$ as below, $$ f(k) = \exp(j 2 ...
2
votes
2answers
224 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + ...
5
votes
4answers
150 views

Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you.
0
votes
1answer
31 views

How do I write this complex number in exponential form? [closed]

$$ -4 - i 16\sqrt{5}$$ Example: I know we can write $-8-i8\sqrt{3}$ as $16e^{i(-2\pi/3 + 2k\pi)}$ where $k = 0,\pm1, \pm2,....$
0
votes
1answer
35 views

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
4
votes
3answers
473 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...