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

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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 ...
1
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1answer
20 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
114 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
votes
1answer
66 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
138 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
22 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
49 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
104 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 ...
1
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3answers
93 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
79 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
58 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.
1
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2answers
52 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
46 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
33 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
votes
2answers
94 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 $a, b, c,$ ...
1
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2answers
92 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 ...
1
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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 $ax^yy^x=by^zz^y=cz^xx^z$ ...
16
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6answers
459 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
108 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 ...
1
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2answers
63 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
20 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
votes
2answers
84 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 ...
0
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2answers
81 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.)
0
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1answer
25 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 ...
0
votes
3answers
71 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
41 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
54 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?
0
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1answer
45 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
37 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
votes
1answer
49 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
1
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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
37 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} - ...
2
votes
2answers
68 views

Solving time needed to travel a given distance, given simulated (not real physics) properties of acceleration of object

For a small personal project I'm looking at travel time of objects in a video game called EVE-Online. I need to calculate time it will take object to travel from stand-still, constantly accelerating, ...
4
votes
3answers
42 views

Alternative definition for Exponential and Logarithmic functions to prove identities (and by extension sin and cos related identities)

Edit : This question is about not about proving identities, but representations that are easier to work with than Taylor series or integral definition for the functions exp or ln functions. Please do ...
9
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2answers
80 views

Prove that $\ln$ and $\exp$ are inverses

If we take the definitions of $\exp$ and $\ln$ as follows: $\exp(x) = {\large\sum\limits_{i=0}^\infty} \dfrac{x^i}{i!}$ $\ln(x) = {\large\int_1^x} \dfrac1t\ dt$ how could we prove that these ...
1
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1answer
36 views

Exponential Generating Function Fun

Given the recurrence relation of $a_n = a_{n-1} + n$, for $n \gt 0$, Where $a_0 = 1$. I know the solution is: $a_n = \frac{1}{2}n^2 + \frac{1}{2}n + 1$. I am not having troubles finding this ...
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0answers
46 views

Proof of the following theorem on Exponential Families

I unfortunately can't find a proof for the following theorem from Statistical Inference by Casella-Berger, Theorem 3.4.2, on exponential families. It says the following: If $X$ is a random ...
1
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1answer
68 views

Exponential of a Jordan block [duplicate]

I am having difficulties with calculating exponential of a Jordan block, I cannot understand the method, can please someone help me, I have an exam on Monday. 'J' is my Jordan matrix and 'P' is my ...
1
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3answers
54 views

How to solve a problem with a variable in both the base and exponent on opposite sides of an equation

I am working on systems of equations in Pre-Calculus, and I presented the teacher a question that I had been wondering for a while. $x^2 = 2^x$ The teacher couldn't figure it out after playing with ...
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1answer
60 views

nth root function

I want to write code for a nth root function, so I need to be sure, that the underlying mathematical function is correct. From another post over at SO, I wrote the following definition: $ \sqrt[x]{y} ...
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1answer
36 views

Limit of expression with increasing exponent

I got stuck trying to evaluate: $$\lim\limits_{n\to \infty}\left(\frac{\sqrt2}{\sqrt2+1}\right) + \left(\frac{\sqrt2}{\sqrt2+1}\right)^2 + \left(\frac{\sqrt2}{\sqrt2+1}\right)^3 + \dots + ...
0
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2answers
40 views

How to determine if $e^{-t}(\cos t +i\sin t)$ is periodic

$x(t) =e^{-t} (\cos t+i\sin t)$ determine $x(t)$ is periodic or nonperiodic and the period if its periodic
0
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4answers
50 views

When $t$ goes to infinity in function $-2 (te^{-t} + e^{-t} )$

How to compute the following limit: $$\lim_{t\to\infty}-2\left(te^{-t}+e^{-t}\right)$$
0
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0answers
27 views

Handling large exponents in a matrix

I have four quantities stemming from a 4th order differential equation. I can represent these as a vector which is a product of a 4X4 matrix $$ M=\left\{v,\frac{\partial v}{\partial x},\frac{\partial ...
1
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2answers
35 views

Integration of $|e^{-(2+j)t}|^2$

The integration of $|e^{-(2+j)t}|^2$ from zero to infinity is $1/4$ when I separate above as $|e^{-2t}|^2 \cdot |e^{-2jt}|^2$ and integrate. $|e^{-2jt}|$ was taken as $1$. But when I integrate the ...
0
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1answer
61 views

Calculating biological growth of doubling cells

For some kinds of experiments, biologists use isolated cells grown in culture. Cells differ significantly in their cell doubling times (one cell dividing into two cells). Plant cells ...
0
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1answer
18 views

Help with graphing an exponential function

I know this is probably a very easy question, but how do you know the difference between an exponential that is decaying at an increasing or decreasing rate? I multiplied $e^{i2\omega}$ by a box and ...