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

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0answers
56 views

Are exponents with a base very close to $1$ (such as $1.0001$) useful in Mathematics?

I was curious if exponents with a base very close to $1$ are ever used in Mathematics and for what applications. For example, when I was in college, my Calculus professor told me that logarithms are ...
0
votes
1answer
39 views

Nature of the range of $e^x$

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
1
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1answer
55 views

This three-variable system of equations seems impossible to solve

$$g = af^b + c$$ $$i = ah^b + c$$ $$k = aj^b + c$$ I want to solve for $a$, $b$, and $c$. $f$, $g$, $h$, $i$, $j$, and $k$ are inputs to the equations, so they don't have to be solved for. Just ...
0
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0answers
15 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
1
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0answers
20 views

I can't plot this function (y = (8*(1-exp^(-800000000*x))) on Scilab. Error 144 [on hold]

I'm a beginner using Scilab, so probably it is a stupid question. Also, I don't know how to change Scilab console messages to English, I'm sorry about that. Here is my code: function [y] = f(x) y ...
2
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1answer
42 views

A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
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0answers
18 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
0
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1answer
23 views

What are the rules being used to compute $\lim\limits_{x\rightarrow \frac{\pi}{2}} (1-\cos x)^{\tan x}$?

I am given $\lim\limits_{x\rightarrow \frac{\pi}{2}} \frac{\ln(1-\cos x)}{\cos x} = -1$ So, $(1-\cos x)^{\tan x} = e^{(\tan x) \ln(1-\cos x)}$ and as $x\rightarrow \frac{\pi}{2}$, we have: $(\tan ...
0
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1answer
21 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
1
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1answer
22 views

Constant raised to the power of an even or odd function

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function. Would $a^{f(x)}$ be an even or odd function? And would $a^{g(x)}$ be an even or odd ...
2
votes
5answers
53 views

Prove that $e^x \ge$ its Maclaurin polynomial with n terms [on hold]

a) show that $e^x \geq 1+x$ for all $x\geq 0$ b) deduce that $e^x \geq 1+x+\frac{1}{2}x^2$ for $x\geq0$ c) use induction to prove that for $x\geq 0, n\in \mathbb{N}$ $$e^x\ge ...
1
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3answers
29 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
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1answer
49 views

Why isn't Euler's formula multivalued?

So it seems that all complex exponential functions are multivalued except for ones with base $e$. Why? Shouldn't all exponentials be multivalued?
-1
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1answer
27 views

An exponential inequality

Assume $a(t)\geq 0$ and $b(t)\geq 0$. i can show the following inequality $\mid e^{-\int_0^ta(s)ds}-e^{-\int_0^tb(s)ds} \mid\leq T\max_{0\leq t \leq T}\mid a(t)-b(t)\mid$ by writing $\mid ...
4
votes
5answers
121 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
0
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0answers
10 views

Exponential decay and logarithmic functions

How do you use experiential decay functions and logarithmic to create a mathematical model to compare the ages of two bones (Bone A and Bone B). When Bone A contains $3$ times the amount of ...
0
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2answers
13 views

Re-arrange the following equation to express $t$ in terms of $V$. [closed]

$$V = 2e^{5t-9}$$ I'm not sure how to solve this.
-1
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0answers
18 views

Integral vanishes as $r \rightarrow \infty$, exponent

I'm integrating $e^{-x^2}$ over the boundary of a triangle and in order to finish my solution I need to show that $$\int_0^r (e^{y^2-r^2-2iry})i dy$$ vanishes as $r$ approaches infinity. Could you ...
0
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2answers
37 views

Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$

Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$. What should I approach first? Please help!
0
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0answers
6 views

Second derivative of Bregman divergence

Suppose I define an exponential family distribution: $$ f(x; \theta) = \exp \left( \langle x, \theta \rangle - h(x) - \psi(\theta)\right) $$ where the log-partition function is: $$ \psi(\theta) = ...
1
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0answers
72 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
1
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2answers
43 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
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votes
1answer
41 views

How do I show that $-2^{n+1} + 2^{n} = -2^{n}$ [closed]

How do I show that $$ -2^{n+1} + 2^{n} = -2^{n} $$
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1answer
22 views

What do you do to simplify this radical expression? [closed]

This is the expression that needs to be simplfied Look at the image. Does the parenthesis have effect on the simplification of this problem?
0
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1answer
26 views

Limit of a sequence, power of one minus an exponential

What is the limit of the following quantity $L \rightarrow \infty$, $$ (1 - \exp(-cL))^{\delta L} $$ for any $c$ and $\delta$ positive constants?
2
votes
4answers
172 views

Intiutive argument that $\exp' = \exp$

Is there any intuitive argument or visual "proof" that $\exp' = \exp$? Suppose you have defined the Euler number $\mathrm{e}$ as limit of the sequence $(a_n)$ where $a_n = \left (1 + \frac{1}{n} ...
0
votes
2answers
59 views

Let$\ x$ be a real number between$\ 0$ and$\ 1$. Is it possible to write$\ e^{x}$ as a function of$\ \Gamma \left(x+1\right)$?

In particular, I'm looking for a relation between$\ e^x$ and$\ \frac{1}{ \Gamma \left(x+1\right) }$, which would be of help for a proof.
0
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2answers
75 views

What is the general solution for $y''e^{-y} =1$? [closed]

how can I find the general solution for an ODE $$y''e^{-y} =1?$$ Thanks.
2
votes
3answers
58 views

l'Hôpital and it's use in derivation

In for example $$\lim_{x\rightarrow 0} \frac{e^{ax} - 1 - ax}{1 - \cos x}$$ We would use l'Hôpital rule and derive it twice to get $a^2$ How do you see this when just looking at the given function, ...
0
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2answers
58 views

How to understand $2^{e^{x}}$?

$2^{e^{x}}$ is an exponent over a exponent. It is confusing. How to understand it? Can I simplify it?
1
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2answers
52 views

When all solutions of $y''+ay'+by=0$ are bounded in R?

Could you please help me solve this problem. Suppose $y''+ay'+by=0$ is differential equation with $a,b$ are real numbers. I need to find conditions when all solutions of this equation are bounded. I ...
1
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4answers
38 views

Criterion to satisfy Rolle's Theorem.

$f(x) = \begin{cases} x^a\log x, & \text{if $x \neq 0$,} \\[2ex] 0, & \text{if $x=0$. } \end{cases} $ What should be the value of $a$ so that f satisfies Rolle's theorem in [0,1] ?? What I ...
1
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2answers
57 views

find $\displaystyle \int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$

find $\displaystyle\int \dfrac{e^{-2x-x^2}}{\left( x+1\right)^2}\hspace{1mm}dx$ If I do Integration by parts, I end up with $\displaystyle\int e^{-2x-x^2}\hspace{1mm}dx$ Which I believe cannot be ...
0
votes
2answers
23 views

Help me solve this exponential function problem…

The temperature of a cooling liquid over time can be modelled by the exponential function $$T(x)=60\left(\frac12\right)^\frac x{30}+20$$ where T(x) is the temperature, in degrees Celsius, and x is ...
2
votes
3answers
94 views

Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system ...
1
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2answers
18 views

Sketch the graph and Determine the domain and range of $h(x)=3+e^{-2x}$.

How do I even start on this? How do I sketch the graph and find the domain and range? I am really lost on how to do this problem! Please walk me through this question!
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2answers
31 views

Natural logarithm problem

I'm kind of confused on how to solve this problem! Any guidance/advice would be appreciated. Thanks! $e^{−9}e^{−2}e^{9}$
2
votes
2answers
110 views

How to solve this equation $x^{2}=2^{x}$?

How to solve this equation $x^{2}=2^{x}, x \in R$ ? por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?
2
votes
5answers
130 views

How to solve this exponential equation? $2^{2x}3^x=4^{3x+1}$.

I haven't been able to find the correct answer to this exponential equation: $$\eqalign{ 2^{2x}3^x&=4^{3x+1}\\ 2^{2x} 3^x &= 2^2 \times 2^x \times 3^x\\ 4^{3x+1} &= 4^3 \times 4^x \times ...
0
votes
2answers
19 views

Integral $\int_{-\infty}^0 e^{(-3i+\omega)t} $

Let's say I am integrating this function: $e^{(-3i+\omega)t}$ from $t=-\infty $ to $t=0$ [Note: $\omega$ is just a constant] The same function could be rewritten in this form(i believe?) : ...
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votes
1answer
66 views

How to Solve $-3^x+617x+1625=0$

can anyone please help me solve this : $$-3^x+617x+1625=0$$ I can't do it analytically. originally the problem was to find intersection point of $$y=1625+617x$$ and $$y=3^x$$ i did the regular ...
3
votes
2answers
54 views

How to find the number of solutions of equation $x^n - a^x = 0$?

I have to find the number of solutions of the equation $x^4 - 5^x = 0$ Since it is only asked to find the number of solutions and not the exact solution, what is the best way to approach such ...
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votes
3answers
39 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
0
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1answer
28 views

How to solve an exponential equation

$8^X + 7 (2^{X+1}) = 7 (4^X) + 8$ $2^{3X} + 7 (2^{X+1}) = 7 (2^{2X}) + 2^3$ $3X = 3$ $X =1$ OR $X+1 = 2X$ $X=1$ BUT answer $X = 0$, or $1$ or $2$ ????
0
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1answer
22 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
0
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1answer
43 views

Simplification of a formula with several exponential functions

I'm trying to work this problem my teacher gave as practice, I have the answers but I'm not sure what I'm doing wrong. I'd be grateful if anyone could help me out. Thank you so much.
0
votes
1answer
39 views

Evaluate the limit of $e^{\pi-\ln \frac{x+4}{-x}}/x$ as $x\to 0$

Evaluate:$$\lim_{x \to \ 0^-}\frac{e^{\pi-\ln \frac{x+4}{-x}}}{x} $$ I tried Hopital's rule, even the Taylor series of the function $e^x$ without success. So how can one solve it?
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votes
2answers
37 views

Pre-Algebra: How to solve this in simple way? [closed]

I wanted to know what's the easiest way to solve this math problem. I've got my tests tomorrow and I would be grateful if you could help me out. $$\frac{5^{a-2b} \cdot 125^{2a-b}}{25^{-a-b}}$$ ...
1
vote
1answer
28 views

Find the time required for an investment to grow to a given amount with compound interest

Find the time required for an investment of 5000 dollars to grow to 7400 dollars at an interest rate of 7.5 percent per year, compounded quarterly. Your answer is t= years. I got to the point ...
-1
votes
2answers
23 views

Solve this problem using logarithms… [closed]

The doubling period of a bacterial population is $10$ minutes. At time $t = 90 \text{ minutes}$, the baterial population was $70000$. With t representing minutes, the formula for the population is ...