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

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19
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4answers
4k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
0
votes
1answer
54 views

How to solve the equation 5x=0.01^x [duplicate]

Hi I recently posted a this question earlier and got some excellent answers but to take it a little further I liked k170's answer however it contained a Lambert W Function in the answer and I was ...
2
votes
3answers
191 views

How to solve 5x=0.01^x

I just want to know how to solve: $$\ 5x=0.01^x$$ I have tried to use logarithms. It would be a huge help if someone could help because no matter what I do $\ x$ always gets stuck in a logarithm. ...
9
votes
1answer
107 views

Are these equations known?

Hello I found two equations that lead to constant e. I wonder if they are known. I think especially first one is most likely known but I couldn't find, it is hard to search google with all these ...
1
vote
1answer
23 views

What is this equal to? : $|A+B|^2$ where $A = P e^{ia}$ and $B = Q e^{ib}$

$A$ and $B$ are two complex numbers: $A = P e^{ia}$ $B = Q e^{ib}$ I would like to know what is this equal to? : $|A+B|^2$ Please also give a small proof if possible.
2
votes
2answers
32 views

Finding the interval after substitution

Given this problem $$8\cdot 3^{\sqrt{x}+\sqrt[4]{x}}+9^{\sqrt[4]{x}+1}\geq 9^{\sqrt{x}}$$ After simplifying I get $8\cdot 3^{\sqrt[4]{x}-\sqrt{x}}+9\cdot 3^{2\sqrt[4]{x}-2\sqrt{x}}\geq 1$ now ...
0
votes
1answer
29 views

Simplify expression with lambert w-Function

I have an expression and i am almost sure what it equals: $ e^{-W_{-1}\left(-\frac{log\left(x\right)}{x}\right)} $ I only need a simplified version of this expression for $x\geq e$. I assume: ...
0
votes
0answers
27 views

Moving Logarithmic function equation plotted on log log paper up or down on the y axis

I'm hitting a stump here. I have a logarithmic function plotted on log log paper so it's a straight line. So let's say I have this entire line plotted out on the log log paper....how would I simply ...
0
votes
1answer
52 views

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined?

If $E(z)= \sum _{n=0 }^{\infty }\frac {z ^n } {n! } $, how is $E(0) $ defined? The exponential function for complex $z $ is defined in Rudin's principles as the power series $ \sum _{n=0 }^{\infty ...
0
votes
2answers
76 views

Solve $2^x=13 \bmod 3^4$

Solve $2^x=13\bmod 3^4$ I know $\log13=30\bmod 3^4$ and $\log16=15 \bmod 3^4 $ I've tried subbing $\log13/\log16$ for $2$ but I am not sure what to do next.
0
votes
3answers
44 views

How to do the derivative when an exponent has an exponent

I am trying to solve an equation that is in the form of $y(x) = (c + x^2)^{x^2}$. Note $c =$ constant My initial thoughts are I need to look into using ln and e to solve this. However what I am ...
0
votes
0answers
13 views

Exponential distribution of a value over a range

If I have a value and I want to exponentially distribute it over a set of points how can I do that so that the sum of each point's value equals the original value? Example: I have a value of 5 and a ...
2
votes
2answers
51 views

Where we have used the condition that $ST=TS$, i.e, commutativity?

definition Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$, $$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$ Proposition If $S$ and $T$ are linear transformations on $\mathbb R^n$ ...
5
votes
3answers
260 views

Doubt in the defn of exponential operator.

definition Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$, $$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$ But in this definition, What they are meaning by the term $A^kt^k$, If I ...
1
vote
3answers
58 views

Integral $\int_0^∞ x^2 \exp(-2kx)$ not defined when putting limits

I have this integral : $\displaystyle\int_0^∞ x^2 \exp(-2kx)\,dx$ now, integrating by parts ($x^2$ as first function and $\exp(-2kx)$ as second), \begin{align} I & = \int_0^∞ x^2 \exp(-2kx)\,dx ...
0
votes
2answers
99 views

Determining how long a body has been dead using the number e

I have recently seen a quote about determining how long a body has been dead: “Dead bodies lose heat exponentially, and therefore e can be used in an appropriate equation to determine how long ...
2
votes
1answer
63 views

Find fundamental matrix of a 2x2 matrix with rank 1

$$ x'(t) = \left[\begin{array}{cccc}0&1\\0&t\end{array}\right]x(t)$$ I am having trouble computing the fundamental matrix. I get: $$ x1(t) = x2(0)*exp(0.5t^2) $$ $$ x2(t) = x2(0)*exp(0.5t^2) ...
4
votes
6answers
121 views

Find $\lim_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$

How to calculate the following limit? $$\lim\limits_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$$
0
votes
2answers
68 views

Complicated Exponential Equation

So I was trying to solve the following equation. I'm fairly good at mathematics so the fact that I have no idea what to do in order to solve this question kind of annoys me. I thought I'd see if ...
0
votes
1answer
18 views

Algorithm for smooth exponential curves

I want to plot an exponential curve between 256 and 0. Using the following equasion, I get the resulting data set. (Please note that I am rounding any decimals down to nearest whole number throughout ...
0
votes
1answer
53 views

Equation for exponential deceleration so the objects “stops” at destination

Let's say my ship's velocity during deceleration phase is given by: v(t) = v0 * exp(-k * t) where v0 is the speed at the time ...
1
vote
2answers
32 views

How to take this exponentials

Given an expansion of a cumulant function as follows: $$ \kappa(t) = \frac{t^2}{2} + \frac{\rho_3 t^3}{6\sqrt{n}} + \frac{\rho_4t^4}{24n} +O\left(\frac{1}{n\sqrt{n}}\right), (*) $$ where ...
2
votes
1answer
81 views

How to calculate the sum $ x + x^2 +…+ x^n$ [closed]

How can I get the result of this sum: $$ x + x^2 +...+ x^n $$
2
votes
4answers
78 views

How do you solve $e^x=ex$?

I mean the answer is really simple. It's one but how do you solve this equation?
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
94 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
39 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
33 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
53 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
42 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
33 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
18 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
29 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
26 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
54 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
180 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
75 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
34 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
57 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
17 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
35 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
355 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. ...