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

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2answers
59 views

What's the point of Euler's number in exponents? [closed]

I want to know why we use $(1+e^{\text{something}})^{-1}$ for artificial intelligence. I know $e$ is just $2.7$. So what? Why $2.7$ and not $3$? Does it have a special property?
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3answers
32 views

Understanding exponential decay

Say I have a variable $x$ that decays over time $t$ as follows: $$ \frac{dx}{dt} = \frac{-x}{\tau}. $$ Solving for $x$, I get \begin{align} x &= \frac{-1}{\tau}\int x dt\\ &=e^{-t/\tau}. ...
2
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1answer
38 views

Finding a Transformation for a Sum of Exponentials

I am looking to see if it is possible to find a transformation $T_i(f(x))$ such that $$T_1\left(e^x+e^{ix}+e^{-x}+e^{-ix}\right)=e^x-ie^{ix}-e^{-x}+ie^{-ix}$$ ...
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2answers
51 views

Matrix exponentials

Let $A(x)$ be a real valued matrix with $x$-dependent coefficients where $x\in \mathbb{R}$. What is the necessary and sufficient condition on $A(x)$ such that the matrix exponential $$\exp( - A(x))$$ ...
2
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1answer
52 views

$n$ complex numbers with modulus $1$

The problem: Let $z_1$,$z_2$,...$z_n$ $(n \geq 3)$ be complex numbers such that $\left| z_1 \right|=\left| z_2 \right|=\ldots=\left| z_n \right|=1$. Then show that the following statements are ...
2
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2answers
79 views

How do I integrate this expression involving exponential and polynomial

I tried a few ways (integral by parts, expanding), but I'm unable to compute this integral. $$\int_0^\infty\frac{\lambda^{n}e^{-\lambda}}{(\lambda + b)^2}\, \text{d}\lambda$$ n >= 0, b >= 0
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1answer
54 views

Prove the Inequality on sequence [closed]

$a_n=(1+\frac{1}{n})^n$ , $b_n=\sum_{k=0}^n \frac{1}{k!}$. Show that $b_n-\frac{3}{2n} < a_n < b_n$.
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1answer
49 views

Finding the hourly growth rate

A species of bacteria doubles in population every 6.5 hours. There were 100 bacteria to start with. What is the hourly growth rate of the bacteria? How many bacteria will there be after a day and a ...
0
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1answer
45 views

exponential equation system without log [duplicate]

How should I solve this equation system without using logarythms,using just a simple method? (E.g. turning it into a quadratic one using t) $$\left(\frac{3}{2}\right)^{x-y} - ...
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6answers
69 views

How to deduce the limit relation $\lim_{x\to0} \frac{e^{cx}-1}{x}=c$

Let $f(x) = e^{cx}$ where $c$ is constant. Show that $f'(0)=c$ and use this to deduce the limit relation $$\lim_{x\to0} \frac{e^{cx}-1}{x}=c$$ Proving $f'(0)=c$ is easy but I'm not sure how the limit ...
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0answers
12 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
2
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0answers
50 views

Solving exp integral in closed form?

I am trying to solve the following integrals: 1) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 m^2})} dxdy $ 2) $\int \int e^{-(\frac{x^2}{2 m^2} +\frac{y^2}{2 n^2})} dxdy $ 3) $\int \int ...
0
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1answer
35 views

Solving Equation for $x$

Solve $(a + \sqrt {a^2 - 1})^{x^2 - 2x} + (a - \sqrt {a^2 - 1})^{x^2 - 2x} - a = 0$ for $x$ , where $a>1$ . My approach is as follows : $(a + \sqrt {a^2 - 1}) (a - \sqrt {a^2 - 1})=1 $ Let $(a + ...
5
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4answers
140 views

Dubious “proof” of $e^x$ derivative?

The proof to which I am referring is amply discussed here: Derivative of exponential function proof, but I remain unconvinced by the answers that pertain to the specific proof discovered by ...
2
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4answers
127 views

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$? [duplicate]

What is $\lim\limits_{n\to {\infty}} (\frac{n}{1+n})^n$. Is it possible to write the function $f(x)=x^n$ and since we know $\frac{n}{1+n}\to 1$, so $f(\frac{n}{1+n})\to 1^n=1$. So the limit it $1$. ...
2
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4answers
48 views

Proving uniform convergence of $(1+\frac{x}{n})^n$ to $e^x$ on compact intervals in the real numbers

My goal is to prove that if $b> a > 0$ are real numbers, then: $\lim_{n \rightarrow \infty} \int_a^b (1 + x/n)^n e^{-x} dx = b-a$. I think the best way to do this is to show that $(1+x/n)^n$ ...
2
votes
1answer
74 views

Show by series definition of exponential function that $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty.$

There are many arguments I have seen using $\ln-$ arguments and other properties of the exponential function to show the existence of this limit $\exp(-x) \rightarrow 0 $ for $x \rightarrow \infty$. ...
0
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1answer
28 views

Is it possible to represent pieces of two functions with one equation?

I'm trying to create a rudimentary weighting system for evaluating how close two numbers are to each other. (This corresponds to string lengths - coding project for work... happy to explain in more ...
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3answers
44 views

Calculating exponential limit

I've been breaking my mind over this one. Find the limit. $\lim\limits_{n \to \infty} (\frac{n^2+3}{n^2+5 n-4})^{2n} $ I know it equals $\frac{1}{e^{10}} $ but can't figure out how to find it. Help? ...
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5answers
109 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
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1answer
36 views

Acceleration: If I know distance, time, and initial velocity, what's acceleration and final velocity?

So I know the Initial Velocity ($V_i$), Time ($t$), and Distance ($d$). I know that $$d = V_it + \frac{1}{2} at^2$$ If I rearrange this, would acceleration $a = \dfrac{2(d - V_it)}{t^2}$ ? Then ...
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1answer
42 views

Finding probability a particle will appear after t seconds (exponential r.v)

Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density with $\lambda=1$ The probability $P(0,T)$ that a particle will appear in the next T ...
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
3
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1answer
100 views

Find monotonic functions going from $0$ to $+\infty$ for $x \in (-\infty,+\infty)$ (similar to $e^x$)

How can we find functions on $\mathbb{R}$ with exponential-like properties, namely: $f(x)$ is infinitely differentiable; $f(x)$ and all its derivatives are monotonic; $f(x)$ and all its derivatives ...
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2answers
31 views

Solving exponential equation (quadratic type)

I fail trying to solve the following equation: $9^x-6^x-2^{2x+1}=0$ Trying to write it as a quadratic equation makes my constant term exponential $(3^x)^2-2^x3^x-2^{2x+1}=0$ How can I solve this ...
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2answers
25 views

Clarification Needed Regarding $\sinh^{-1}(-3)$

As the definition of $\sinh^{-1}(x)$ goes : $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^{2}+1}\right)$ So what I expect to get is $\sinh^{-1}(-3)=\ln\left(-3+\sqrt{10}\right)$ The value inside of the ...
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1answer
26 views

Using exponential decay function to predict outcome

Let's say I have a graph that follows the function $y= ae^{-bx}$ , and I'm trying to predict the chlorine residue left in a pool after a certain amount of time. So for $2$ hours, the chlorine residue ...
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0answers
51 views

Is there a way to solve the exponential equation $a^x + b^x + c^x = d$ analytically?

So I came across this equation. $$a^x + b^x + c^x = d$$ where $a, b, c$ and $d$ are all constants. And I just wondered, is there any way to solve for x analytically?
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3answers
34 views

Is there a non-exponential function whose limit at infinity is a real, irrational number?

$e$, for example, can be calculated through a non-polynomial function $(1+1/x)^x$, but I cant think of an example for a non-exponential function (or rational function) where the limit to infinity ...
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1answer
63 views

Prove that $n! = O(n^n)$

I thought $n^n$ was greater than $n!$. How would I go about proving this? I have this so far: Assume that $P$($n$) is true $n!$ = O($n^n$) Assume that $P$($n+1$) is also true $(n+1)! ...
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1answer
65 views

Two problems with Exponents

How to solve following problems on exponents: $$\frac1{1+p^{a-b}+p^{a-c}}+\frac1{1+p^{b-c}+p^{b-a}}+\frac1{1+p^{c-a}+p^{c-b}}=?$$ and If $a^2bc^2=5^3$ and $ab^2=5^6$, what is $abc$? Please ...
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1answer
49 views

How to solve this difficult one variable equation analytically?

Would anybody like to explain me clearly how to solve analytically this equation? $$5.56=\frac{1-e^{-5.5x}}{1-e^{-x}}$$ I have already solved it with Mathematica and it gives $x=-0.004809$. However, ...
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0answers
28 views

calculating growth rate with exponentials vs. non-exponentials?

i'd like to calculate the growth rate of growing bacterial population. at $t_1 = 0 hr$, the population size (based on density) is: $N_1 = 0.17$ and at $t_2 = 0.5hr$ it is $N_2 = 0.25$. using a simple ...
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2answers
35 views

How can I solve this trig exponential equation ??

I tried taking $\log_2$ of both sides and I got: $(\sin(x))^2\cdot(1-\cos(x))=1$ Is it correct? If it is, how should I continue ?
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2answers
52 views

$\lim_{n\to \infty} (1+n+\cos n) ^\frac{1}{2n+n \sin n}$

While in class, we were proving a limit problem using the Squeeze Theorem, but when I was reviewing my notes, I came up with a problem,, The first question was to prove that $$\lim_{n\to ...
2
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2answers
68 views

Does this definition of $e$ even make sense?

This sprung from a conversation here. In Stewart's Calculus textbook, he defined $e$ as the unique solution to $\lim\limits_{h\to 0}\frac{x^h-1}{h}=1$. Ahmed asked how do you define $x^h$ is not by ...
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2answers
44 views

What is $\frac{(-2)^{x}}{2^{x-1}}$

The title says it all: $$\frac{(-2)^{x}}{2^{x-1}}$$ How is this computed? I'm reviewing the finer points of exponents so a thorough explanation would be most appreciated!
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1answer
27 views

Can anyone walk me through calculating the differential equation…

I need help solving this equation, i've attempted using numerous methods. But I'm give choices with square roots as an exponent of e, and I haven't been able to match any of them. $$ ...
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1answer
33 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
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2answers
15 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
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1answer
34 views

Probability of exponential growth event

Under the assumption of exponential growth of a population of cells, the population size at time $t$, $N(t)$, is: $$N(t) = N_0\exp(rt)$$ where $r$ is the rate of division and $t$ is time. What is ...
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1answer
16 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
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0answers
14 views

Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
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1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
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0answers
44 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
0
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0answers
20 views

Proving Exponential Convergence

Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all ...
0
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0answers
26 views

Closed form roots of sum of exponential functions

Do anyone know a way to solve an equation like the following (over the complex numbers)? $1+2^z+3^z=0$ I certainly cannot. I've tried by hand, and by mathematica, but I can't figure it out. Thanks in ...
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1answer
29 views

Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
0
votes
3answers
54 views

Solve for $x$ for the following exponential equation $2^{2x+1} = 3^{2x+1}$. What am I doing wrong?

$2^{2x+1} = 3^{2x+1}$ $2^1=3$? Why can't I take $\log_2$ of both sides ?
2
votes
1answer
52 views

Exponential equation on the set of real numbers

Solve the following equation on the set of real numbers: $8^x+27^x+2·30^x+54^x+60^x=12^x+18^x+20^x+24^x+45^x+90^x$ $x=1; x=0; x=-1$ are trivial solutions, but I'm stuck with proving that there are ...