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

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0
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0answers
27 views

A question on function [on hold]

Given $g(y) = \ln(y)$ and $f(y) = \frac{\ln(y^2 + 2y + 1)}{2}$. Show that $g(y) - f(y) < 0$ for $> 0$. What does the question mean by "for $> 0$"?
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0answers
25 views

Is this function commonly known or has some name?

I have used this function for fitting in my research, and I wonder if there is a name for it, or is it commonly known in some reduced form? $f(x)=\alpha\frac{e^{-\gamma x}}{x^\beta}+\delta$ Actual ...
1
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2answers
40 views

How to approximate $x^y$ using a quadratic function

I need to build an algorithm that finds the approximately $x^y$ where $x = [0, 1]$ and $y = [0, 0.4)$. This is for a computer algorithm (the standard library is too slow). I thought about making a ...
10
votes
0answers
162 views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
0
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0answers
35 views

Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. ...
1
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1answer
59 views

Exponential of 4x4 matrix

It is asked to calculate $e^{tA}$, where $$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 ...
-3
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0answers
25 views

clarification domain and range of functions [on hold]

Want a clarification on the domains and ranges of addition, subtraction, multiplication, division and enhancement of functions, if possible with an example with roots and absolute value for a ...
-1
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0answers
97 views

Finding the value of $(x+y)$ when $x^x+y^y=31$

As the title suggests...the question is to find the value of $(x+y)$ when $x^x+y^y=31$.Is it possible to solve this question without trial-and-error method when only this much information given.Using ...
2
votes
6answers
70 views

Derivation for the derivative of $a^{t}$ from The Equation

In Calculus, the Equation is known as: $$f'(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ This equation allow us to find the derivatives of functions. Let's try this with the exponential ...
0
votes
3answers
21 views

Comparison of two values

I have to figure out the relation between the quantity $(0.9/1.1)^2 +(1.1/0.9)^2$ and 2. How can i do this without explicitly calculating the first value, by using some laws of exponents?
0
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0answers
6 views

How to test if an number generated from an exponentially distributed random variables is statistically different from zero at some confidence level?

For instance, let the generated number be 0.3 and the mean of the exponential distribution where this number comes from be 0.03. At a significance level of 0.10 or 0.05, is it 0.3 statistically ...
2
votes
1answer
39 views

Function of single variable $f(x)$, $f(x+y)=f(xy)$ and the exponential.

For a function of a single variable $f(x)$ which has the property that $f(x+y)=f(x)f(y)$ we first set $x=y=0$ and then develop an ODE to show that $f(x)=\exp\{-\beta x\}$. I do not understand how ...
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2answers
85 views

Solving $x^{2n} = \frac{1}{2^n}$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
-1
votes
3answers
41 views

Natural Logarithm Question Help [closed]

Would like (and appreciate) some help with this question. If you can please show full working out, thank you. The question asks: Simplify by expressing as a single natural logarithm.
2
votes
2answers
39 views

One integral involving integrals exponential and logarithmic function

Is there a closed-form solution for the integral $$ \int_{0}^{\infty}\log_{2}(1+ax)\cdot e^{-bx} \; \mathrm dx $$ with $a, b \geq 0$? If there is no closed-form solution, whether there is an ...
-1
votes
1answer
21 views

Math - Exponential Function (Help) [closed]

I'm getting lost with all the working out and my results don't seem to fit the criteria. I need to 'Simplify, and express in terms of positive indices' the expression $$\frac {\sqrt{81x^3} - ...
0
votes
1answer
37 views

Simplifying an expression with exponents

I've found this exercise in my work book and I'm scratching my head trying to figure this thing out, a step by step guide and answer would be really helpful! :-) Simplify, and express in terms of ...
0
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0answers
22 views

An incorrect proof that the Lie algebra matrix exponential is always injective. What's wrong?

Suppose we have two square complex matrices $X,Y $ in the lie algebra $\mathcal{G}$ of matrix Lie group $\mathbb G$ such that $e^X = e^Y$. Then $e^{tX}$ and $e^{tY}$ define the same one-parameter ...
1
vote
1answer
52 views

Why is $\frac{1}{x} \sum_{n=1}^x \ln (n) \sim \ln(x) - \gamma$

I was playing with some functions and decided I wanted to see at which point the factorial of $x$ became bigger than $e^x$. I set them equal to each other and after doing some algebra I ended up with ...
0
votes
0answers
38 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
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2answers
31 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
0
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0answers
55 views

how did he turn from that from to that from ? is that possible or wrong mathematically [closed]

$$U=e^{x+\ln x}= (e^x)(e^{\ln x})= (\ln x)e^x$$ How did he turn it from $(e^x )(e^{\ln x})$ to $(\ln x ) e^x$
1
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1answer
43 views

Is the following solvable for x?

I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base: $[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt ...
1
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2answers
28 views

Function that fulfils the equation

Prove that for all $x\in\mathbb{R}$ there exists only one $y=y(x)$ that fulfils the equation: $$3x+e^x=y+e^y$$ I am completely lost with that. What should i do?
2
votes
0answers
36 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ ...
0
votes
0answers
29 views

How to prove $x^m = O(e^x)$ for any $m \gt 0$?

My attempt: It's true for $m = 1$ clearly. Now assume true for $m=1\dots M-1$. Then $x = O(e^x)$ and $x^{M-1} = O(e^{M-1})$. Lemma: if $f = O(g)$ and $f' = O(g')$ then $ff' = O(gg')$. Proof: $f = ...
1
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1answer
19 views

The Limit of an Integral Containing Exponentials

I am unsure how to show this. Suppose $\delta(s)$ defined on $(-\infty , s_*)$ is increasing and satisfies $\lim _{s\rightarrow s_*} \delta = \lim _{s\rightarrow s_*} \frac{d \delta}{d s} = \infty$ ...
4
votes
1answer
76 views

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$

How to show that $|\exp(z)-1|\le2|z|$ for $|z|\le 1$ ...
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0answers
19 views

solve implicit equation with lambertw, exponentials, logarithms and first order polynom

I have a very complicated problem to solve. I am almost sure it's impossible to solve but maybe one of you guys has a miracle solution for me. I am modelling the behaviour of a photovoltaic cell and ...
1
vote
3answers
53 views

Exponential integral representation

According to exponential integral eqn. (8) $\; E_{1}(x) \;$ can be represented by: $$ E_1(x)= - \gamma - \ln(x) - \sum _{n=1}^{\infty } \frac{(-1)^n x^n}{n n!} $$ where $\gamma$ is the ...
0
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1answer
47 views

Evaluating the integral $\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt$

It is stated that (for $\lambda>0$) $$\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt = 1-\frac{|\mu|}{\lambda}$$ for $ 0\leq|\mu|\leq\lambda$, and zero ...
0
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1answer
39 views

Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula

I have a problem deriving the following formula: $$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$ Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
1
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1answer
46 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
0
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2answers
22 views

Trying to understand an expansion/limit from geometric sum to exponentials, what kind of rule is at play?

Can someone help me understand what's going on here? This is for a problem involving moment generating functions, which is related to statistics and probability, but I figured it was more of a math ...
2
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0answers
50 views

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? [duplicate]

Does every even degree polynomial in the expansion of $\exp x$ have no real roots? I tried the first few values and it seems like it... Is this a known result?
2
votes
0answers
80 views

Tough integral with exp

Can anybody integrate this: $$ \int_0^K e^{i(A\sqrt{\mathstrut k^2+m^2} - Bk)} dk,$$ where $K$, $A$, $B$ and $m$ are real constants? Sorry folks, I didn't realise anybody would be interested in the ...
4
votes
2answers
152 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
3
votes
4answers
59 views

Using the $\ln(\cdot)$ for $(1-e^{-x})$

The given function: $$B= A(1-(e^{-x}))$$ Now, I want to 'destroy' the e-function by taking the logarithm of it. First, since $\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + ...
2
votes
2answers
44 views

What is the fallacy of this proof?

I recently was working with square roots and came across this- $({\sqrt -1})$$=-1^\frac12$$=-1^\frac24$$=(-1^2)^\frac14$$=1^\frac14$$=1$ I understand that this is not true,but despite repeated ...
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0answers
26 views

How to solve the integral in this case?

If I have a Kernel from the path integral technique, and I want check if the product rule is valid in 2D, (I know it is) then I need to solve the following integral: \begin{equation} \text{Let} \quad ...
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1answer
18 views

exponential behavior from pattern of data

In the image below from this video lesson, the teacher shows how to get an exponential function from a pattern of data, also copied below. You can see that her solution using the formula (a)(b) to the ...
7
votes
5answers
379 views

Limit of a function involving a sequence.

I have the following problem: Suppose that $\lim_{n \to \infty} a_n = 0$. Prove that for any $x$ $$\lim_{n \to \infty} \left(1+a_n \frac{x}{n}\right)^n = 1.$$ I have tried replacing $a_n$ with ...
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votes
1answer
47 views

If $(t-2)= e^{3(x-1)}$ then $x=?$ [closed]

If $(t-2)= e^{3(x-1)} $ then $x=?$. I guess I have to change the right side of the equation to get the x to the other side.
6
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2answers
135 views
0
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1answer
24 views

I want to find PDF by differentiating CDF and then from PDF, expected values of the following problem. .

Three light bulbs have independent exponentially distributed lifetimes with a common parameter $\lambda$. What is the probability distributed function and expected value of the time until the last ...
4
votes
2answers
97 views

What is intresting about $\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}$?

Why does $$\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}=\frac{1}{\ln{x}}$$ There only seems to be a relation when using square roots, but not for cubed roots or ...
1
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2answers
29 views

Exponential functions with negative base

Consider the function $f(x) = (-2)^x$, $x$ belongs to irrationals. For which $x$ does $f(x)$ belong to the reals.
5
votes
3answers
133 views

Simplifying integral:$\int_{0}^{\infty}{\exp\left(-\left(u^2+{ {\alpha^2}\over {16u^2t}}\right)\right)}~\mathrm{d}u$

$$I(t)=\int_{0}^{\infty}{\exp\left(-\left(u^2+{ {\alpha^2}\over {16u^2t}}\right)\right)}~\mathrm{d}u $$ where $\alpha$ and $t$ are positive constant. P.S.I would like to edit this problem, because ...
7
votes
4answers
458 views

Exponential Simultaneous Equations

Solve the following simultaneous equations: $$2^x + 2^y = 10$$ $$x + y = 4$$ Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved ...
0
votes
2answers
38 views

How to calculate matrix exponential of a $2\times 2$ matrix with repeated e values

Specifically, I am trying to calculate the matrix exponential, $e^{At}$, where A = $\begin{bmatrix}-1 & 1\\-9 & 5\end{bmatrix}$. I calculated the the E values to be 2 with a multiplicity of 2 ...