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

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1answer
27 views

Combinatorics problem using generating exp functions 2

Calculate the number of sequences of length n that are made of $1, 2, 3, 4$ so that the digits $1,2$ shows an even number of times, And the digit $3$ shows at least 1 time. I've been given a clue to ...
1
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1answer
50 views

Combinatorics problem using generating functions 1

In how many ways can you divide $n$ different balls into $5$ different boxes so that the two last boxes has an even number of balls. I've been given a clue to show that: $\sum_{n=0}^\infty {x^{2n} ...
2
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3answers
89 views

Why does the minimum value of $x^x$ equal $1/e$?

The graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for ...
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2answers
15 views

describing a decay process with exponentials and differential equations

I have a process of degradation of some material that proceeds like this across time $t$: $C_t = C_{t-1} + RC_{t-1}$ where $C_t$ is the amount of material at time $t$ and $R$ is a (negative) rate of ...
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2answers
33 views

Algebra of exponential

Solve for $x$ in exact value: $\\3^{2x}-3^{x+2}+8=0$ I have tried substituting $3^x$ $=a$ but I didn't get anywhere. $\\a^2-a^{1+\frac{2}{x}}+8=0$
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2answers
37 views

Solving exponent on both side of equation

I'm new here on Mathematics and have only basic algebraic knowledge. I have a problem in how to solve the following equation: $$ P^x = R_0^x + R_1^x + ... +R_n^x $$ I know the value of P and the ...
0
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1answer
48 views

exponential martingale inequality

I stumbled across a claim I couldn't verify. Let $M_t$ be a continuous local martingale, $M_0=0$ a.s. and $\lambda>0$. Then $$ \mathbb{E}\left( \exp \left( \lambda M_t \right) \right) \leq ...
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3answers
33 views

Transformation of exponentials

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried: Let $y=3^x$ to $y=e^{x'}$ $$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$ ...
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0answers
55 views

How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...
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2answers
86 views

Solve exponential equation $3^{x-1}+5^{x-1}=34$

What should I do? If we divide say with $3^{x-1}$, we win nothing, considering the $34$. How do we solve this equation? Thanks.
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1answer
12 views

Creating an exponential function limited to a range

I'm trying as part of an algorithm to make a function so as to adjust dynamically a parameter of the algorithm (the cooldown rate) based on the current temperature (again in terms of the algorithm). ...
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4answers
65 views

Simultaneous Equations (Stuck on the algebra)

Question: Solve the following simultaneous equations for real values of x and y $$ \left\{ \begin{array}{l} 9^{2x+y} - 9^x \times 3^y = 6 \\ \log_{x+1}(y+3) + \log_{x+1}(y+x+4) = 3 ...
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2answers
45 views

How do I solve $x^{k}=k^{x}$, given k, analytically? [duplicate]

I've tried taking the log of both sides (but am left with $k\ln{x}=x\ln{k}$).
2
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1answer
26 views

Properties of the per-element exponential (Hadamard exponential) for matrices

I'm asking this question mostly out of curiosity, though I do also have a potential application. In linear algebra we usually define the matrix emponential as $e^A = I + A + \frac{1}{2}A^2 + ...
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2answers
53 views

Prove $e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X}$ if $\hat{X}^{2}=I$.

If we have an operator $\hat{X}$ such that $\hat{X}^{2}=I$ (the identity), how do we prove that: $$e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X} \ ?$$
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2answers
30 views

Simplifying an exponential expression.

maybe this is a stupid question but I have the following expression: $ 10^{-18}(e^{50,9702078⋅0,75}) = 10^{-18}(4⋅10^{16}) $ How would I go about simplifying the big exponent on the left to what's ...
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1answer
19 views

Conditions on the real part of an exponential function

If $a = \mu +i \omega$, what conditions are necessary to impose on $\mu$ and $\omega$ if $Re(e^{at})$ for $t>0$ is to be: a) exponential decreasing b) exponential increasing c) oscillating with ...
5
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3answers
113 views

simple proof that $\sqrt{1+\frac{1}{x+1/2}}(1+1/x)^x\le e$

It is well known that for $x>0$ that $\left(1+\frac{1}{x}\right)^x\le e\le\left(1+\frac{1}{x}\right)^{x+1}$ (see wikipedia). However, one can obtain the stronger inequality $$ ...
3
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1answer
63 views

Finding inverse function of $f(x)=10^x+5^x+1$

As the title of the question says , how to find $f^{-1}$ for this example ? Of course $f$ is one-to one and with a simple transform it would be $y-1=5^x(2^x+1)$ $\Rightarrow ...
5
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2answers
132 views

The general solution of $x^a = a^x$ for real $a >0$

What are the roots of $$f(x) = x^a - a^x$$ for real $a > 0$? Case 1: For $0 < a < 1$ there is 1 solution, $x=a$. Case 2: For $1\le a < e$ there are 2 solutions: $x=a$ and $[x>a]$. ...
0
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1answer
19 views

Generate exponential weights (sum of all = 1)

I have $500$ observations and I want to make exponential weighted average of them. I want the weights to be something like $w_i = 0.999^t$ when $t$ is from $1$ to $500$ (num of observations). ...
2
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3answers
48 views

Why does $\lim_{n\to \infty} (1+\frac{1}{n!})^{2n} = \lim_{n\to \infty}\big( (1+\frac{1}{n!})^{n!}\big)^{\frac2{(n-1)!}}=e^0$?

I understand the algebraic manipulation and I'm assuming that the thinking is: $\lim_{n\to \infty}\left( \left(1+\frac{1}{n!}\right)^{n!}\right) = e$ and $\lim_{n \to \infty} \frac2{(n-1)!} = 0$. ...
0
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2answers
102 views

Why does $\lim_{x \to \infty} \big(1 + \frac{1}{x}\big)^x = \lim_{x \to 0} \big(1 + x\big)^{\frac{1}{x}}$?

Is there a way to make sense of that relationship? Could you derive one from the other algebraically? It looks like the first limit approaches $e$ from lower values (for positive $x$ values) whereas ...
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3answers
91 views

What am I missing in this argument for $\lim\limits_{x\rightarrow \infty} \ln x = \infty$?

In an appendix of Stewart's Calculus, the logarithmic and exponential functions are built up starting from the defnition $\ln x = \int_1^x \frac{1}{t}\,dt$. Having shown that $\ln(x^n) = n \ln(x)$ ...
2
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2answers
74 views

What does $e^{-iAt}$ mean?

I'm trying to understand an algorithm, to solve $Ax = b$ linear equations. But there is an equation, which I can not understand: $e^{-iAt}$ What does it mean, to calculate the $-iAt$ power of $e$ ?
0
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1answer
55 views

a simple inequality for exponential functions

Ho to prove the following inequality: $e^{x}\leq1+x+x^2$ for $|x|\leq1/2$. It looks simple, but I don't know where to start.
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2answers
49 views

Does an infinite iteration of a function still have my solution and why does it work?

I found the other day that you could find some solutions to an equation in the form $$f[f(x)]=x$$As a matter of fact, I found some solutions to$$f[f(f(\cdots f(x)\cdots))]=x$$ The solution, if one ...
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2answers
45 views

Factoring things out of a trig function? New trig. identity???

I noticed the other day that$$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$ Allow $x=ab$. $$\cos(ab)=\frac{e^{iab}+e^{-iab}}2$$ Where upon you can use some exponential identities to ...
0
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1answer
31 views

Which one is exponential rate of growth: doubling at each step or time-squared? [closed]

Doubling I mean - like the penny thought experiment; that doubles each day. Sometimes one is being given as an example, and sometimes the other. Time-squared is probably the exponential rate of ...
4
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2answers
89 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
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2answers
84 views

How to simplify a sum of exponential equation?

Suppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ...
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1answer
34 views

Determining the value of parameters given constraints

If $$\frac{x(y+z-x)}{\log x}=\frac{y(z+x-y)}{\log y}=\frac{z(x+y-z)}{\log z}$$ and $$ax^yy^x=by^zz^y=cz^xz^y$$ then what is the value of $a + \frac b c$? I am getting as $ax^yy^x=by^zz^y=cz^xx^z$ ...
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6answers
386 views

Prove that $e>2$ geometrically.

Q: Prove that $e>2$ geometrically. Attempt: I only know one formal definition of $e$ that is $\lim_\limits{n\to\infty} (1+\frac{1}{n})^n=e$. I could somehow understand that this is somehow related ...
6
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1answer
102 views

Solution to $xe^{e^x}$

The problem $xe^{e^x}=e$ came up another day and I wondered if it were solvable. My attempt was the following substitution,$$x=W(u)$$$$W(u)e^{e^{W(u)}}=e$$Where I used a Lambert W identity to get ...
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2answers
61 views

How to calculate limit of series

I have many limits for homework that I dont know how to solve them. I tried many things, but dont have any idea. Hope you can help me $$\lim_{n\to \infty} n*c^n $$ when $$\lvert c\rvert < 1$$ ...
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2answers
40 views

How to use the properties of the logarithmic function

I'm coding the game asteroids. I want to make a levels manager who can create a infinity number of level increasing in difficulty. My levels have as parameters : The number of asteroids on the ...
0
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1answer
19 views

exponential growth calculated in two ways

Maybe quite basic question, but was little surprise for me. Lets say we start with $2$ units (maybe thousands of microbes) and we have $30 \%$ increase (growth rate) over time unit. The question is ...
2
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2answers
80 views

How to solve this equation algebraically?

I've come across this interesting equation which I do not know how to solve. The equation is: $$e^x+\log x =1$$ I used WolframAlpha to solve it and it got but, it didn't provide any solutions. The ...
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2answers
75 views

Solve for $t$: $ e^{-2t} + 2t = 4 $

How do we do this problem for other values of the constant, say 300 or -1000? Is there a general way to solve such questions? (Looking for a way to solve this with pen and paper.)
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1answer
22 views

Is it possible to represent a root as a simple rational function with an exponent?

Using the following function:$$y=\frac{mx^p+b}{d}$$... where $m$, $p$, and $b$ may be any integer ... where $d$ may be any integer $\gt0$ ... and where $x$ may be any rational number $\ge0$ Is it ...
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3answers
65 views

How to find the PDF for Y=e$^x$

How do I find the PDF for $Y$ = e$^X$ when $X$ is $N$(μ,σ$^2$) I have seen the problem where $X$ is $N$(0,1), but I am curious on how to find it given just these parameters?
2
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1answer
39 views

A function and its derivative chasing tails

For which $t\ge0$ does there exist a differentiable function $f$ with $f(0)=0$, $f'(x)>f(x)$ for all $x>0$ and with $f'(0)=t$? This question was inspired by (and is a variation of) the ...
2
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1answer
51 views

$3^x-2^y=1$, $x \in \mathbb{N}$ and $y \in \mathbb{N}$

$3^x-2^y=1$ or $y=\log_2{\left(3^x-1\right)}$ $x$ and $y$ must be natural numbers. I know this two solutions: $x=1$ and $y=1$ $x=2$ and $y=3$ Are there more solutions? How can I find them?
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1answer
42 views

Proving a Hermite polynomial equality

For $$H_{k}(x)=\frac{(-1)^{k}}{\sqrt{k!}}\exp\left\{\frac{x^{2}}{2}\right\}\frac{d^{k}}{dx^{k}}\exp\left\{-\frac{x^{2}}{2}\right\}$$ I want to prove $H'_{k}(x)=\sqrt{k}H_{k-1}(x)$. So far I have ...
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1answer
32 views

Exponential equation Exercises

Today I have big problem. Our teacher gave us this HW with exponential equation for marks. I do not want to get bad mark so I am here. Please help me. I did a lot. And I also know how to solve basic ...
3
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1answer
46 views

Verify proof of $ f(x)=e^x $ if $ f(x+y)=f(x)f(y) $ and $ f'(x)$ exists for all $x$

This is exercise 6.26.8 from Tom Apostol's Calculus I, I'd like to ask someone to verify my proof. I'd be also interested in alternative proofs: If $ f(x+y)=f(x)f(y) $ for all $ x $ and $ y $ and if ...
0
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1answer
28 views

$EXP$ formula in excel using $e$

Can someone explain what this formula is doing? $$=EXP(x)/(1+EXP(x))*100$$ If you are not familiar with $EXP$, $EXP$ calculates the $e^x$. Thanks
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4answers
41 views

Exponential function taking away a constant each day

Suppose someone has a lake with an area of $A = 1240~m^2$ in this example). It is covered by an area of $c = 10~m^2$) of algae at the beginning ($t = 0$), which doubles each day. This gives us the ...
0
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0answers
35 views

Critical points of $(x^2+y^2)\exp(y^2-x^2)$

I have been given the following exercise as homework: "Find the critical points of the function $$f(x,y)=(x^2+y^2)\exp(y^2-x^2)$$ and determine whether they are maxima, minima or saddle points." So, ...
0
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2answers
23 views

The set of real solutions to an equation

$$4^x - 7(2^{\frac{x-3}{2}}) = 2^{-x}$$ Set of real solutions is in which interval: $(-9, -2)$ $(0, 3]$ $(-2, 0]$ $(7, 12]$ $(3, 7]$ I tried the following. Dividing by $2^{-x}$ I get $2^{3x} - ...