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

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1answer
31 views

Proving properties about complex exponential

I defined $a^z$ for $z \in \mathbb{C}$ as $a^z = \exp(z\log(a))$ and I proved it is continous, now I want to show that $a^n = a \cdot a \cdot a \cdot \ldots \cdot a$ for $n \in \mathbb{N}$ so ...
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1answer
43 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
2
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1answer
88 views

Problem of the Week! [duplicate]

This week in Algebra II we are studying the Hanoi tower's. Our assignment was to find what type of formula would give the number of moves it would take to solve the puzzle. After using a T-chart ...
0
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1answer
61 views

Derivative of exponential functions

Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves? i.e. $$ y = e^{x^2}\\ ...
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4answers
97 views

Limit of $\frac{e^{x}}{\ln(x)}$

I don't know how to find the limit $$\lim_{x\to\infty}\frac{e^x}{\log x}.$$ How can I do this ? Thank you in advance.
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1answer
30 views

Show that $g(x)=x\ln{x}$ and $g(x)=e^x$ are bounded below.

Show that $g(x)$ is bounded below, for $0\leq x$: a) $g(x) = \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \\ x\ln{x} & \mbox{if } x>0 \end{array} \right.$ b) $g(x)=e^x$ For (a), ...
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1answer
77 views

Prove logarithm rules using definition as the inverse exponential

Problem $3$. Show that $\operatorname{exp} : \Bbb R \to (0, \infty)$ is bijective. Its inverse function is called the (natural) logarithm $\log : (0, \infty) \to \Bbb R$. Verify the logarithm ...
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3answers
429 views

“Linearize” an exponential-looking graph with log function

This may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize" ...
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1answer
99 views

Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. ...
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1answer
72 views

Relation/connection between $n!$ or $e$ and $2^n$

What is the relation/connection between $n!$ or $e$ and $2^n$ ? Is the there a relation/connection between $n!$ or $e$ and $2^n$?
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0answers
15 views

How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
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2answers
159 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
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3answers
39 views

Analytical aptitude - Division of exponentials.

What is the remainder when 6^17 + 117^6 is divided by 7? How to approach these type of questions?
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1answer
78 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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1answer
73 views

When does this non linear 2 equation system have solutions? What is the solution?

I need to solve the following system: $$ \begin{cases} a x_0^2 = \exp{ \left( -\dfrac{x_0^2}{4 \sigma^2} \right) } +a r^2 \\ \exp{ \left( -\dfrac{x_0^2}{4 \sigma^2} \right) } + 4 a \sigma^2 = 0 ...
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2answers
28 views

Calculus Exponential Functions Again

This one wants us to evaluate the following limits of this exponential function. $$\lim_{x \to \infty} \frac6{e^x-6}$$ I'm not sure how to approach this problem. I did easily figure out this version ...
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1answer
96 views

(Basic High School Mathematics) Graphing the inverse square law

I did an experiment measuring the intensity of light in relation to the distance away from a source. How would I graph the avg intensity over 1/distance squared? Note that T1 = trial 1 etc.. It's ...
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1answer
54 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
2
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7answers
129 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
0
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3answers
116 views

$0^0$ — indeterminate, or $1$? [duplicate]

One of my teachers argued today that 0^0 = 1. However, WolframAlpha, intuition(?) and various other sources say otherwise... 0^0 doesn't really "mean" anything.. can anyone clear this up with some ...
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1answer
86 views

How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get ...
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1answer
31 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
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4answers
71 views

$f:\mathbb R \to (0,\infty)$ defined by $f(x)=e^x$. Describe its inverse.

How do I go about describing it? Well first is the inverse $e^{-x}$ or $\ln(x)$? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... $f:\mathbb R\to ...
3
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1answer
61 views

Proof of $ e^x-x^2 \gt 1 $ when $ x \gt 0$ and $x$ is a real number .

I want to Prove $ e^x-x^2 \gt 1 $ when $ x \gt 0$ and $x$ is a real number . For this purpose , my trying is as the following : $ e^x-x^2 = \{1+x+\dfrac{x^2}{2!} + +\dfrac{x^3}{3!}++\dfrac{x^4}{4!}+ ...
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1answer
34 views

Rearranging the terms so that the denominator becomes the numerator

I have the equation $$ \frac{120}{1 + 3.167 \cdot e^{-0.05t}} = 60 $$ How do I transform it so that the denominator becomes the numerator? This would make the problem much easier.
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1answer
42 views

Convergence rate of exponential function

If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can ...
2
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1answer
67 views

How to solve $6^{2x}-10\cdot 6^x=-21$ using logarithms?

What do I do with $\large 6^{2x}-10\cdot 6^x=-21$? Since $6$ and $-60$ are not of the same base (nor can they be written as exponents of the same base cleanly) I am having trouble solving for ...
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2answers
40 views

Derivatives of Logarithmic functions

I am stuck in these problems. $\displaystyle \frac{d}{dx} (\log_2 x^8)$ $\displaystyle \frac{d}{dx} (e^x \ln x)$ I think for the first problem the answer is $\dfrac{2}{x^7}$, whereas for the ...
3
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1answer
163 views

How can we describe the graph of $x^x$ for negative values?

We usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line: This graph may seem like ...
2
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1answer
35 views

How to simplify $\cot(\sec^{-1}(e^x))$

I've been trying to simplify $\cot(\sec^{-1}(e^x))$. I thought substitution might the way to go about it so I said: let $u = \sec^{-1}({e^x})$ I'm therefore trying to find $\cot(u)$ From $u = ...
2
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1answer
47 views

Second order linear ODE not making sense…

I am given: $y''-3y'+2y=0$ $y(0)=1$ $y'(0)=2$ I know that $r_1=2$ and $r_2=1$ The solution therefore is: $y(x)=C_1e^x+C_2e^{2x}$ Solving for initial values, I have: $y(0)=C_1+C_2=1$ ...
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1answer
54 views

How to solve this system of ODE's?

I'm not sure how to proceed to solve this system of ODE's; $$ \begin{bmatrix}\dot{x}_1 \\\dot{x}_2\end{bmatrix}=\begin{bmatrix} \cos t & -\sin t\\ \sin t & \cos t ...
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3answers
713 views

How to solve exponential function inequality?

How do I solve the exponential equations like $2^\frac{x}{8}<x$? I can solve this by plotting into graph. But is there any way to do it mathematically? or like $2^x < 100x^2$ . I am trying to ...
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2answers
55 views

$\sup\{a^{r}\mid r<x; r\in\mathbb{Q}\}=\inf\{a^{s}\mid x<s; s\in\mathbb{Q}\}$ How to prove it?

This proposition is a lemma related to another stage for defining exponential function $a^{x}$, in this case for reals, taking into account it is defined for rationals. Proposition Let $a>1$ and ...
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0answers
17 views

relationship between two set of variable

i am trying to determine what kind of mathematical modeling could be applied following two variables,let us call them $x$ and $y$ ,namely change one variable has effect second on,i have several ...
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2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
0
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1answer
86 views

Solve exponential equation $(1−ax^2)∗e^{bx^2}=c$ for x. transcendental algebraic equation?

is it possible to solve an equation with the given form analytically? $$ (1-ax^2)*e^{bx^2}=c $$ $$ e^{bx^2}-ax^2e^{bx^2}=c $$ I've already tried it using a logarithmic function but I cannot manage ...
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3answers
114 views

Exponential Regression: how to get a formula from a given pattern

I'm trying to code a computer script (in Java) that returns an array of numbers that follows a certain pattern. The numbers should be: ...
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1answer
61 views

Creating an exponential scale

Good morning, I am trying to create an exponential scale for attributing values in a scoring model. Here is the function I was thinking of using: y = z^x Where: y = Score X = Risk assessment ...
0
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1answer
31 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
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1answer
94 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
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2answers
32 views

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $ for $d,k \ge 0$ I know that $(1+\frac{1}{n})^n \le e$ but does that help? Actually, I don't really 'know' this, but I've heard it's true at least ...
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2answers
61 views

finite sum over a Gaussian

I have a sum of the form: $$\sum_{n,m=-N}^N e^{-\alpha (n-m)^2}$$ where $\alpha > 0$ is some constant, and $n,m$ take the integer values: $-N,..,N$. I know there is a possibility of exchanging ...
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1answer
43 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
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3answers
135 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
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1answer
36 views

Just one step away. Exponential formula

The data points are. I have worked out that. y =.032(2.5)^x is correct where my table is x = 0, y = .032 x = 1, y = .08 x = 2, y = .2 How do I get my formula to reflect y=-3, x =.0320 ...
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0answers
40 views

Sigmoid Function Question

Ive been trying for well over a week to try to understand how to use a simple sigmoid or logistic function works. Specifically I'm trying to understand how to build proper polynomia parameters for ...
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2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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2answers
305 views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
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4answers
117 views

Solve equation: $5^x = -2x + 7$

How to solve that equation: $$5^x = -2x + 7$$ I already have the answer $x=1$. Can anyone please explain to me?