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

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2
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1answer
33 views

What is The Output of $(a^x*b^{-x/2})(a^{x-y}*b^y)$?

I was solving some problems when I came across a "slick" way to solve these types of problems, and I substituted different numbers for the variables and the answer has something to do with ...
0
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0answers
23 views

Solving an exponential equation in like terms

This one may be fairly easy, yet, for the life of me, I can't remember how to do it. I would like to solve this equation to express x in terms of ...
0
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1answer
11 views

Limit of complex exponential

The following is the characteristic function of a random variable $X_n$:$$\phi_{n}(t)=\frac{1-e^{it}e^{\frac{it}{n}}}{(n+1)\left(1-e^{\frac{it}{n}}\right)}$$ for $t \in \mathbb R$. I am trying to ...
5
votes
2answers
81 views

Proving uniqueness of $e$ [duplicate]

Let's define $e$ as the number $a$ such that $\frac {d}{dx} a^x = a^x$. I'm trying to prove that this $a$ has to be $e$. I don't see any way of proceeding from here except by the limit definition ...
2
votes
1answer
24 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
2
votes
1answer
55 views

How can I solve for x where $10^{10000} = x^x$

I hope this is not too elementary a question to post on here. If so, apologies. I'm stumped how I would solve for x where $10^{10000} = x^x$. Thanks!
4
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0answers
46 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
0
votes
2answers
30 views

Simple Explanation needed

I have been struggling with basics lately, so for this problem is-The value of $e^{-\infty}$ is 0 because $\frac{1}{e^{\infty}}=\frac{1}{\infty}=0$. Am I right ?
2
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1answer
31 views

Uniform convergence to exponential exercise

Yesterday I encountered the following exercise in a tutorial sheet from the University of Lyon : define a sequence of functions $(f_n)$ (with $f_n:[0,\infty) \to {\mathbb R}$) by ...
7
votes
2answers
74 views

How does $\int_0^\infty e^{-t^4}dt = \Gamma (\frac{5}{4}) ?$

My text book claims that $$\int_0^\infty e^{-t^4}dt = \Gamma \left(\frac{5}{4}\right).$$ I fail to see this. By the definition of the gamma function we have $$\Gamma (z) = \int_0^\infty ...
0
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2answers
30 views

How can I evaluate this exponential equation with natural logarithm $6161.859 = 22000\cdot(1.025^n-1)$?

I'm trying to evaluate an exponential equation with natural logarithm, but I'm certainly doing something wrong, can someone explain me how would you solve it using natural logarithm? $$6161.859 = ...
1
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2answers
44 views

Find 2 imaginary numbers that have a cosine of 4, using $\cos z =\frac{e^{iz}+e^{-iz}}{2}$

Use the definition $$ \cos z =\frac{e^{iz}+e^{-iz}}{2} $$ to find $2$ imaginary numbers having a cosine of $4$. I tried two approaches, both of which ended in failure: $$ 8=e^{iz}+e^{-iz}\\ ...
1
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0answers
18 views

Math equation problem - fitting wallpapers on a wall

I am building up a java program but don't have the right idea on how to resolve its math problem. The tasks I am doing are: I have to cover the wall with wallpaper. The wall is "a"(input) meters ...
2
votes
1answer
60 views

On Properties of Exponentially Prime Numbers

A usual prime number is a number greater than $1$ which is not in the form of multiplication of two numbers greater than $1$. We may consider the following natural generalizations: $p>1$ is $+$ - ...
1
vote
1answer
35 views

How can I write in Landau notation (or the like) that $2^x/x$ rises almost as fast as $2^x$?

Since $2^x \not\in O(2^x/x)$, we do not have $O(2^x/x)=O(2^x)$. But since $x$ rises linearly and $2^x$ exponentially, $2^x/x$ rises almost as fast as $2^x$. Can I somehow express this in Landau ...
0
votes
1answer
20 views

Exponential ( Problem with the mathematical description)

Well, I know the mathematical description of e as $\lim_{n\to\infty}(1-\frac{m}{n})^{n}=e^{-m}$ ,but today in my statistical mechanics class, while calculating the volume of thin shell of thickness ...
-1
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3answers
46 views

evaluate $\lim(1+3x)^{(1/2x)}$.

Evaluate $\lim(1+3x)^{(1/2x)}$ as x approches 0 We know that : $\lim(1+x)^{(1/x)} =e$ "as x approaches 0 " I know that we can manipulate this equation to get $e$ to some power and I tried so many ...
0
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0answers
26 views

Exponential distribution - Expected length of time

This was one of the test question that my teacher had put up. There is an electric board which contains 10000 bulbs, time out burnout for each bulb is an independent random variable following ...
0
votes
1answer
20 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...
0
votes
2answers
19 views

x increases with 125% each week, what is x week n?

x has the value 1. How would I calculate the accumulative value of x for a certain week, if ...
0
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2answers
27 views

Solving an exponential equation with logoarithm laws

$$2(5)^x = 3^{x+1}$$ I am trying to solve for $x$ in the above equation. Is there a way to make the bases the same to solve? Can I simplify the left side to $10^x$? I'm really not sure where to start ...
2
votes
2answers
32 views

Solve the following exponential equation

$$7^{3x+1}=5^x$$ I am trying to solve this equation. I solved the equation and got what I believe to be the correct answer, but when I verify the answer it appears to be incorrect. Any idea why? Here ...
1
vote
1answer
16 views

Amount of Pharmaceutical Left in the Patient's Body After the nth Dose is Administered

A patient is administered 500 mg of a certain pharmaceutical every 6 hours. The half-life of the pharmaceutical in the body is 130 minutes. Determine the amount P(n) of the pharmaceutical that remains ...
5
votes
2answers
65 views

How to compute $(e^x -1)/x$ when $-10^{-8}<x<10^{-8}$

I am trying to graph the function $y=(e^x -1)/x$ for values of $x$ close to zero. The result should be $y$ close to $1$. Howevere, both in Excel and WolframAlpha the calculated values become very ...
2
votes
2answers
23 views

Simple explanation needed for exponential

While going through my professor's notes while calculating integral with branch cut, I came across this relation.It's basic,I guess.So,How this relation come from ...
1
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1answer
21 views

Exponential algebra problem: Equating powers

Given Data: 5 a = 26 125 b = 676 What is the relation between a and b? I simplified 125 b = 5 4 + 5 a but how to equate a and b in above relation? Note: Answer should be in the format xa = yb ...
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4answers
27 views

Derivative of a square root with exponential function

So I have the following function: $f(x)= \sqrt{e^{2x}}$ After applying the chain rule I sit with: $$\frac{1}{2\sqrt{e^{2x}}}2e^{2x}$$ From there I got: $$\frac{e^{2x}}{\sqrt{e^{2x}}}$$ While the ...
0
votes
1answer
37 views

Integrating an exponential function

How do I integrating this function- I=$\int \exp{\mathrm{\frac{(-1)(z^2-k)^2}{2}}}dz\\$ limit is from -infinity to +infinity I tried a lot by nothing worker!Pls help!
1
vote
1answer
31 views

Why does an improper integral turn into an answer with factorial?

Suppose I have $\int_{0}^{\infty}y^{2n+1}e^{-y}dy$ Why does this integral equal $(2n+1)!$ ? Could somebody please explain this?
0
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0answers
28 views

Approximate Sum of exponentials

I want to closely approximate the following sum into possibly a single term (not as infinite summation) $SUM= \exp^{(-a_1x_1-a_2x_2-a_3x_3)} - \exp^{(-v)}\exp^{(-b_1x_1-b_2x_2-b_3x_3)}$ here $a_i, ...
1
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0answers
31 views

Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
3
votes
3answers
26 views

Proof of transpose property of matrix exponential

Using the fact that the matrix transpose distributes over infinite sums to show that $e^{(A^T)} = (e^A)^T$. I feel like this is really trivial, but I don't know quite how to prove this. How would I ...
2
votes
1answer
76 views

Logarithm of Matrix exponential

Can we say $\log(e^X e^Y)=X+Y \tag 1$ ? where $X$,$Y$ are general skew symmetric matrices of order $3 \times 3$ (Just mentioned skew symmetric matrices to indicate that these are rotational ...
1
vote
1answer
55 views

Math C question related to diseases. [closed]

There are only 4 people in my math C class, including my teacher. We were given the question below and asked to find the answer, unfortunately we all have different answers. What is the answer to the ...
1
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1answer
35 views

exponential generating function for bernoulli numbers [closed]

How I can find exponential generating function for this sequence $(2^n − 1) B_n,$ where $B_n$ is Bernoulli numbers
0
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1answer
48 views

Does the matrix exponential take open sets into open sets?

This is from Hall's Lie Groups, Lie Algebras, and Representations, in theorem $2.13$: Let $B_\varepsilon$ be the open ball of radius $\varepsilon$ about zero in $M_n (\mathbb{C})$ [$= ...
2
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3answers
36 views

Exponential growth and decay

There's a cup of coffee made with boiling water standing at room where room temperature is $20ºC$. If $H(t)$ is the temperature of this cup of coffee at the time $t$, in minutes, explain what the ...
0
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0answers
18 views

The probability of a function whose components obey exponential distribution

Thanks for your attention, here are the details: $${x_i} \sim \exp \left( {{\lambda _i}} \right) , i = 1,2, \cdots ,6 ,a > 0$$ $$\Pr \left\{ {1 + \frac{{a\frac{{{x_1}}}{{{x_3}}} \cdot ...
0
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2answers
25 views

Conversion algorithm needed.

I just started an online course in math, and have a problem with an exercise. I don't know what this type of math is called in English, so please forgive me for the bad title (and tags). The ...
1
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2answers
49 views

How to solve $(x-1)e^{-x} > 0.5$

As the title mentioned, how to solve $x$ from the equation: $$(x-1)e^{-x} > 0.5$$ How can I solve this analytically? This is a part of my homework and I got stuck to this equation. I'm also ...
1
vote
3answers
68 views

Implicit differentiation with e

I am trying to find $\frac{dy}{dx}$ of $$e^{2y}+2e^x = 3$$ I am able to get as far as differentiating both sides of the equation, but then I struggle in the algebra to solve for y. Can someone hold my ...
1
vote
1answer
51 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
1
vote
1answer
21 views

Asymptotic relation between specific binomial coefficient and exponential function

I need to determine the asymptotic relationship between the functions: $$f_1(n)={n\choose{\lfloor{n\over{2}}\rfloor}}, f_2(n)=7^{\sqrt{n}}$$ (I'm going to just assume $n$ is always even.) I've ...
2
votes
3answers
33 views

Derivative of an exponential function

I am trying to solve $$\frac{1}{e^{x}}$$ I first tried using the quotient rule, and ended up with: $$\frac{e^{x}}{(e^{x})^2}$$ That was not the right answer, so I took a look at wolfram, and ...
0
votes
1answer
38 views

An integral involves Gamma function

Thanks for your attention, I meet an integral involves Gamma function and exponential function as follows:$$\int_a^\infty {{x^\alpha }} {e^{cx}}\Gamma \left( {s,bx} \right)dx$$ where $a > 0,s ...
0
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0answers
85 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
3
votes
1answer
86 views

Prove that the exponential function is differentiable

Imagine that you are writing a book on the foundations of analysis. You have already proved that for each $a > 1$ there is a unique function $f_a(x) = a^x$ satisfying the following: $f_a$ is an ...
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3answers
34 views

How to solve $n$ from $c \leq 1.618^{n+1} -(-0.618)^{n+1}$

I need to solve the bound for $n$ from this inequality: $$c \leq 1.618^{n+1} -(-0.618)^{n+1},$$ where $c$ is some known constant value. How can I solve this? At first I was going to take the ...
0
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1answer
34 views

Challenging Question: for Expected Value of a particular probability density function

I've been stuck on this for a while and it's been driving me crazy. Any help would be greatly appreciated. I am trying to find the Expected Value of the following Probability Density Functions (where ...
0
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1answer
21 views

Differential equation: the law of natural growth and the law of natural decay

I understand that $\frac{dy}{dx} = k*y$ and when $k>0$ this is the law of natural growth and when $k<0%$ this is the law of natural decay, but my textbook gives an example of radioactive decay ...