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Questions tagged [exponential-function]

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

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Population growth precalculus, exponential growth

A population growth model is given by the following formula $P(t)=Ae^{rt}$, where $P$ is the population after $t$ years, $A$ equals the initial population $(t=0)$, and $r$ equals the annual growth ...
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1answer
44 views

How to calculate the integral of exponential functions?

Having an integral like $\int_{2}^{10}{\frac{x}{\ln x}}dx$ How does this function turns to an exponential integral of the form: $ \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,$ For ...
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13 views

Expressing power law decay in terms of exponentials

I'm trying to find out how power law decays can be represented, or approximated, by exponential functions. Any papers or textbook suggestions would be particularly helpful. But in particular, on the ...
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1answer
26 views

How to find arc length (exponential function)

I am sorry I don't know how to use the MathJax equations format properly. I have to find arc length of $ y=( e^{x/2} + e^{-x/2} ) $ over this interval [-2,2]. I found the derivative of y and ...
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3answers
170 views

Exponential/Logarithmic equation system

Solve the following equation system over the real numbers $$\begin{cases} x(1-\log_{10}(5))=\log_{10}(11-3^y)\\ \log_{10}(35-4^x)=y\log_{10}(9) \\ \end{cases} $$ For the functions in the above ...
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3answers
28 views

Proof by Induction help (inequality)

I need to prove if, $$a_1=1,\ a_2=2$$ and $$a_n=2a_{n-1}+a_{n-2}$$ then $$a_n\leq \left(\frac{5}{2} \right)^{n-1}$$ Proof (by using strong induction): As far as I go, is that I prove both bases ...
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33 views

Integral involving complex exponential function

I'm trying to solve the following integral $$\int_c^{\infty}\frac{e^{-\alpha(u+i\,\pi/2)}\,e^{t\, e^{u+i\,\pi/2}}}{(u+i\,\pi/2)^{\beta+1}}du-\int_c^{\infty}\frac{e^{-\alpha(u-i\,\pi/2)}\,e^{t\, e^{u-...
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2answers
62 views

How many roots does an exponential polynomial have?

Let $s$ be a complex variable and consider two polynomials with real coefficients: $$A(s) = s^n + a_{n-1}s^{n-1}+\ldots+a_1s+a_0,$$ $$B(s) = s^m + b_{m-1}s^{m-1}+\ldots+b_1s+b_0,$$ where $n \ge m$. ...
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2answers
52 views

How do I factorise $729^x-243^x-81^x+9^x+3^x-1$?

How do I factorise $$729^x-243^x-81^x+9^x+3^x-1 ?$$ I know the answer is $(3^x-1)(9^x-1)(27^x-1)$, but I have no idea how to go about it. This is actually a step of another question in my book. Would ...
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Solve exponential equation 17 . 64^(x) - 4 . 8^(x) - 164 = - 42 . 8^(1-x) + 2 . 8^(2-x)

I tried to solve this exponential equation but got only one of the two solutions right. This is what I did: 17 . 64^{x} - 4 . 8^{x} - 164 = - 42 . 8^{1-x} + 2 . 8^{2-x} I set everything to one side:...
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2answers
23 views

Integral involving Gamma distribution

I need some help with an integral. This is the solution to one of the problems I had to do. Everything is fine, but I don't understand one step: Now how is $$\int_0^\infty \frac{\beta_n^{\alpha_n+k}}...
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Derive all the Present values and Future values formulas assuming continuous compounding (for multiple cash flows)

$\lim_{n\rightarrow \infty } A(1+\frac{r}{m})^{mn}=Ae^{rn}$ Derive all the Present values and Future values formulas assuming continuous compounding (for multiple cash flows). Can anyone help out? ...
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23 views

Describing curves of complex valued functions

I wish to describe the curves $|f|$=constant and arg$f$=constant for the following functions: 1.$f(z)=exp(z^2)$ 2.$f(z)=exp\left(\cfrac{z+1}{z-1}\right)$ My thoughts: I can write down what the ...
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3answers
23 views

Solving steps for equation with exponent and addition (x^2 + x) = 2y

I have this equation: ($x^2$ + x) = 2y Which I know solves to: x = (-1 + sqrt(1 + 8y)) / 2 x = (-1 - sqrt(1 + 8y)) / 2 However, I have no idea about the steps to reach the solved equation, any ...
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2answers
32 views

equation with exponential functions 2

Solve the following equation over the real numbers: $$ (3+ \sqrt{5})^x + (3- \sqrt{5})^x=7 * 2^x $$
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2answers
59 views

equation with exponential functions

Solve the following equation over the real numbers(preferably without using calculus): $$ 4^x + 4^{1/x} =18 $$ I already know the solutions thanks to Wolfram, what I have trouble with is proving ...
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1answer
40 views

Clever equalities proven similarly to Euler's Identity

From How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, a very elegant proof of Euler's Identity was given. Namely, observing $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, ...
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2answers
33 views

Complex number vs Complex exponential

I know that a complex number is a point in 2D plane. I wonder how to describe what is a complex exponential?
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1answer
68 views

Show that $e^{-1/x^2}$ is not analytic around $x=0$.

I have been working on the following question. Define a function \begin{align*} f(x)= \begin{cases} e^{-1/x^2}&\text{ for }x>0,\\ 0&\text{ for }x=0 \end{cases} \end{align*} Prove that $f$ ...
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3answers
62 views

What is the formula for this given function? [closed]

How do we calculate the formula for the following function? $$ f(n,k) = \frac{1}{n} + \frac{(n-1)}{n^2} + \frac{(n-1)^2}{n^2} + \frac{(n-1)^2}{n^3} + \frac{(n-1)^3}{n^3} + \frac{(n-1)^3}{n^4} + ... + ...
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12 views

Complex exponential series with different limits

Can anyone help in beginning the expansion to this? I know what the general expansion of a complex exponential series should look like and what Cn is equal to but because the limits to this function ...
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2answers
19 views

Find extrema and monotonicity intervals of $ x^{x^2} $

I'm trying to find minimum, maximum and intervals of monotonicity of following function: $ f(x) = x^{x^2} $ I tried calculating derivatives, but they get very, very complicated really fast. How to ...
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1answer
25 views

f(x) greater than g(x) when x > 2 explanation

Where $f(x) = (x^x)^x$ and $g(x) = x^{(x^x)}$. How do I go about showing that one of these functions is always greater than the other for all $x > 2$? Could I use induction? I've gone through it ...
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1answer
27 views

Integrate the solution of a the Matrix differential equation

I have: $\dot{\textbf{x}}=A{\textbf{x}}$ where A is a nxn matrix. This equation has solution: $\textbf{x}(t)=e^{\textbf{A}t}\textbf{x}(0)$ A book i'm reading states that: $\textbf{x}(t_{2})=e^{\...
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2answers
31 views

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $ (The second sum in the left-hand side of the inequality is an exponent) I've ...
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1answer
50 views

Can someone help me with this proof? Some exponential stuff

Let $$x=10^{1/(1-\log z)}$$ and $$y=10^{1/(1-\log x)}.$$ Show that $$z= 10^{1/(1-\log y)}.$$ Don't forget the appropriate assumptions. ($\log z$ is logarithm with base $10$ and argument $z$.)
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2answers
38 views

Top Base Fractional Number

For example: $ 2^3 = 2 \cdot 2 \cdot2 $ But what about a fractional power such as: $$ 2^{2/3} = ? $$ 1) How would I explain this? 2) How would I find value? 3) How would computers calculate ...
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28 views

Two Equations and Two Unknowns

$$\frac { - \left( 2 e ^ { h + l } \left( 2 ( - c + l - 1 ) e ^ { h + l + l } + ( n - 2 ) ( - c + l - 1 ) e ^ { h + l } + n ( - c + l - 1 ) e ^ { l + q + l } - 2 e ^ { h + l } \right) \right) } { \...
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1answer
34 views

Proof $(e^x-1)^r = \sum_{n=0}^{\infty} \sum_{k=0}^r (-1)^k \binom{r}{k} (r-k)^n \frac{x^n}{n!}$

In our combinatorics script I found this $$(e^x-1)^r = \sum_{n=0}^{\infty} \sum_{k=0}^r (-1)^k \binom{r}{k} (r-k)^n \frac{x^n}{n!}$$ I tried finding a proof for this on the stackexchange math and ...
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23 views

Is there a way to compute the exponential of a PDP-1 matrix?

I am computing the exponential of a matrix via Taylor expansion to prove the end-result with induction. For a matrix $A=PDP^{-1}$ where $D$ is the diagonalized matrix, is there any kind of formula ...
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0answers
20 views

Possible description/definition of exponential function (proof verification/review)

I wish to describe the exponential function as unique continuous group isomorphism $f:(\mathbb{R},+)\rightarrow(\mathbb{R}^+,\cdot)$ satisfying $f(1)=a$, $f\equiv a^x$. Lemma 1: Assume $f:G\...
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1answer
63 views

Does my definition of an exponential-like function make any sense?

The definition of the exponential function is based on an infinite series $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} $$ To make things more complicated, we could replace the factorial with the Gamma ...
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1answer
36 views

Find the image of the line $y=mx$ under the exponential application.

Find the image of the line $y=mx$ under the exponential application. I'm not sure if this question will make sense since I've translated it from french to English, but I'm not sure how to start this.....
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1answer
18 views

Understanding proof that exponential map of compact connected Lie group is surjective

Let $G$ a compact connected Lie group. Then, the exponential map $\exp: LG \rightarrow G$ is surjective. (where $LG$ is the Lie Algebra of $G$). $\textbf{Proof:}$ For any torus $T' \subset G$ we have ...
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1answer
35 views

Positive solution for an exponential equation

Define a function of $t$, $F(t) = e^{X_1t}-e^{X_2t}-e^{X_3t}+e^{X_4t},$ for some fixed real values $X_1, X_2, X_3, X_4 \in \mathbb{R}$ and $X_1<X_2 < X_3 < X_4$. Whether $F(t)$ has at most ...
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6answers
57 views

How come $e^x-e^{-x}=0$ does not have a solution? [closed]

While solving a partial differential equation following this document, they state that $$e^{\sigma L}-e^{-\sigma L}=0$$ does not have a solution and ask why. Here $L$ is a constant and $\sigma$ is a ...
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1answer
87 views

How to solve this DE involving a convolution $\int_0^t ds \ e^{-\kappa s} \cos(\omega s) f(t-s)$?

I've got a DE of the form $$ \frac{df}{dt} = A - B f(t) - C \int_0^t ds\ e^{-\kappa s} \cos(\omega s) f(t-s) $$ which I want to solve given an initial condition $f(0^{+})=f_0 \in \mathbb{R}$. All the ...
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2answers
35 views

Is Euler's formula valid for complex arguments

I found this question here : Evaluate $\cos(z)$, given that $z = i \log(2+\sqrt{3})$ It says that - $$e^{-iz} = \cos(z) - i \sin(z)$$ isn't necessarily true because $$\sin z$$ is imaginary (for the ...
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4answers
66 views

Analytically determine if $f(x) = f'(x)$ is possible?

I was taking a test and two true/false type questions were asked. In one of them, I had to say if there is a function $f(x)$ such that $f(x) = f'(x)$. Of course, $e^x$ is such a function and almost ...
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1answer
27 views

Domain of validity of an inequality involving exponential and logarithms

$$x^{{2(\log(x)})^3-1.5\log(x)} \geq 10^{1/2}$$ where $\log$ is decimal logarithm. Just solve it and that's it. Can someone help me with some ideas or even with a solution? I have no idea how to ...
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1answer
46 views

$\lim_{n \to \infty}\frac{e^\sqrt{n}}{c^n}$?

How could I show that the following limit tends to $0$? The constant $c$ can take any value strictly greater than $1.$ $$\lim_{n \to \infty}\dfrac{e^\sqrt{n}}{c^n}$$ I'm having trouble on how to ...
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1answer
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how to express a number only in exponential form of product of 3 and 7 or say any two numbers?

please guide me how to expand a number in the form of $3^x 7^y$ example: $1 = 3^0 7^0$ $21 = 3^1 7^1$ i have tried by finding numbers which can be expressed in form of any one such as for only $3^...
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3answers
42 views

Prove $e^x$ is its own derivative via power series?

$$ \frac{d}{dx}e^x =\frac{d}{dx} \sum_{n=0}^{ \infty} \frac{x^n}{n!}$$ $$ \sum_{n=0}^{ \infty} \frac{nx^{n-1}}{n!}$$ $$ \sum_{n=0}^{ \infty} \frac{x^{n-1}}{(n-1)!}$$ This isn't as straightforward as I ...
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3answers
31 views

Question about Exponential Series Definition and Convergence

I've seen in many textbooks that the following is just a definition: $$e^x = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$$ And then many textbooks just go ahead to prove the absolute convergence of the ...
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0answers
22 views

Find the coordinates of the stationary points of $e^x +ye^{-x} = 2e^2$.

Find the coordinates of the stationary points of $e^x +ye^{-x} = 2e^2$. So I have differentiated this implicitly, which I think is correct but I'm then unsure how I'd actually solve the equation. $$\...
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3answers
43 views

Expression for $e^1$

I am currently looking at "Calculus" by Michael Spivak and in his proof for the irrationality of $e$ he writes the following "We know that, for any n, $e=e^1=1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{...
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2answers
126 views

A conjecture about power sum : $e^{ab}+e^{bc}+e^{ca}\geq 3e^{\sqrt{abc}}$ and $a+b+c=3$

Hello I have this to propose : Let $a,b,c$ be real positive numbers such that $a+b+c=3$ then we have : $$e^{ab}+e^{bc}+e^{ca}\geq 3e^{\sqrt{abc}}$$ For a generalization I have this conjecture : ...
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1answer
38 views

Poisson Variable with an Exponential Parameter becoming a Geometric Distribution?

Suppose Λ ∼ exponential(γ) and X ∼ Poisson(Λ). Use moment generating functions to show that X + 1 ∼ geometric(p) and determine p in terms of γ. In order to solve this problem, I first did: $E[e^{s(X+...
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1answer
23 views

A uses 100nlog(n) operations while B uses n^(1.5) operations. Determine the value n0 such that A is better than B for n ≥ n0 assuming log base 2

I've tried a few different approaches but I'm not getting anywhere with this. 100log(n) = n^(0.5) ==> log(n) = n/100 ==> n = 2^(n/100) Stuck at this dead end. 100log(n) = n^(0.5) ==> log(n^(100)) =...
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0answers
19 views

Solving for $t$ in $\left|\alpha\sin\left(2\pi x\cdot t-\frac{\pi}{2}\right)\right|-y=\left(\alpha-y\right)\cdot\exp\left(-\frac{t}{p}\right)$

Well, I've the following problem: solve (exact) the following equation for $t$: $$\left|\alpha\sin\left(2\pi x\cdot t-\frac{\pi}{2}\right)\right|-y=\left(\alpha-y\right)\cdot\exp\left(-\frac{t}{p}\...