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

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0
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1answer
48 views

Determine value $b$ in $f(x)=ab^x$ given the following data points [closed]

If $f(x)=ab^x$, what is the value of $b$ if $(0,35)$ and $(3,125)$ are data points? Is this the way to do it? $$35=ab^0,$$ $$a=35.$$ $$125=ab^3,$$ $$125=3\log(35)+\log(b),$$ ...
0
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0answers
31 views

Best goodness-of-fit measure to determine: is my dataset power law or exponentially distributed?

I would like to determine whether a discrete dataset that I'm modeling would be better fitted by an exponential function or a power law function. I'm aware that a chi-squared test may be a suitable ...
1
vote
1answer
34 views

Simplifying the Pauli matrix expression $e^{-i\sigma_x\phi/2}$

As in the title, the expression is: $$e^{-i \sigma_x \phi/2}$$ Where $\sigma_x$ is: $$\left\{ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right\}$$ Where would I even begin in simplifying ...
1
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1answer
48 views

$\exp(i \theta)=1?$

So I was thinking, $\exp(i\theta) = \exp( i\theta\cdot2\pi\cdot\frac{1}{2\pi})$, we can rearrange it, so that: \begin{align} & \exp\left( i\theta\cdot2\pi\cdot\frac{1}{2\pi}\right)=\exp\left(2\pi ...
1
vote
3answers
76 views

How to prove this? $ \lim_{x \to 0}\frac{e^x-1}{x}=1 $ [duplicate]

Any idea how do I prove the following? $$ \lim_{x \to 0}\frac{e^x-1}{x}=1 $$ Thanks
1
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4answers
81 views

How to determine the monthly interest rate from an annual interest rate

I have a calculation which gives me the annual interest rate if I already know the monthly interest rate as follows: (Monthly interest rate + 1)^12 In this case I ...
2
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0answers
32 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
1
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1answer
45 views

Solving natural logarithms with absolute value

Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
0
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1answer
25 views

Simplification involving exponents to base e

I've found the following expression. It looks really simple - so it's driving me crazy, that I don' get it: $(e^{3x}).(2)$ is simplified as $2e^{2x}$. Similarly, $(2x+7).(3e^{3x})$ is simplified as ...
1
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4answers
51 views

Prove $f'(z)=f(z)$ implies $f(z)=ce^z$ for some $c \in \mathbb{C}$ [duplicate]

Suppose that: $$f'(z)=f(z) \text{ for all }z\in\mathbb C.$$ In other words, the complex function $f$ is equal to its own derivative. Prove that there is a constant $c\in\mathbb C$ such that $f(z)=c ...
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0answers
48 views

Exponential fields as structures with three binary operations.

The exponential rings and fields are usually studied as structures with two binary operations $(+,\cdot)$ and one unary operation $\exp(x)$ defined on a set $K$. Why not consider the exponential as a ...
2
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2answers
72 views

Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime ...
1
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1answer
43 views

What is the remainder of $|e-\sum_{j=0}^n{1\over j!}|$?

I have to find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$, but I want to do it with the remainder. I know that it is ${e^c\over (n+1)!}$ where $0<c<1$, but how do you get ...
2
votes
1answer
74 views

System of exponential equations

If $x,y,z \in \mathbb{R}$ and $$ \begin{cases} 2^x+3^y=5^z \\ 2^y+3^z=5^x \\ 2^z+3^x=5^y \end{cases} $$ does it imply that $x=y=z=1$?
0
votes
1answer
26 views

Half life, exponential decaying equation question

If a radioactive substance has a half-life of $10$ days, in how many days will $1/8$ of the initial amount be present? Assume the decaying process is continuous (exponential). Will the answer just be ...
5
votes
4answers
876 views

Euler's formula, is this true? [duplicate]

*I've changed this question as below. Let me have a function such as $ f(k) = \exp(j 2 \pi k ) $, where $k$ is real value. Using Euler's formula, we can write $f(k)$ as below, $$ f(k) = \exp(j 2 ...
2
votes
2answers
202 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + ...
5
votes
4answers
148 views

Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you.
0
votes
1answer
30 views

How do I write this complex number in exponential form? [closed]

$$ -4 - i 16\sqrt{5}$$ Example: I know we can write $-8-i8\sqrt{3}$ as $16e^{i(-2\pi/3 + 2k\pi)}$ where $k = 0,\pm1, \pm2,....$
0
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1answer
34 views

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function?

Is $Y=aX^b\cdot\exp(X)$ a rational or exponential function? $Y$ and $X$ are real variables, $a$ and $b$ are parameters. Someone said this is a product of polynomial and exponential function. Do we ...
4
votes
3answers
365 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
2
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0answers
38 views

How to estimate parameters of exponential functions?

I am a new bird to math! I have an expression as follows, $$e^{-ux}+e^{-uy}=z$$ I have at least ten pairs of $(x,y,z)$, so how can I estimate the value of $u$? And how to evaluate my results? Hand ...
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2answers
83 views

Solve $e = xe^x$

I know it it seems trivial that $x = 1$, but I would like to know a more rigorous solution involving algebra. I tried solving for it, but could not come up with a proper solution. My attempt: $e = ...
1
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0answers
19 views

Solve $b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x$

Suppose, \begin{align*} b_1 e^{-a_1x^2}-b_2 e^{-a_2x^2}-b_3 e^{-a_3x^2}=0, \forall x \end{align*} Assume $a_1,a_2,a_3, b_1,b_2, b_3>0$ What are the possible values of $a_1,a_2,a_3, b_1,b_2, ...
5
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2answers
84 views

the value of $e$ and the method of getting it

We define e to be a number which satisfies the following condition $$\lim _{a \to 0} \frac{e^a-1}{a}=1. $$ How did we arrive to the following from above equation $$e=\lim _{n \to \infty} ...
1
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2answers
49 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
6
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4answers
1k views

Slick proof of exponential inequality

Today I saw that using taylor series, one can show that $e^x+e^{-x}\leq 2e^{x^2/2}$. Is there a slick proof using some sort of Jensen-type inequality or integral bound?
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0answers
50 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
1
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3answers
45 views

Need help with an inverse function

$$g(x) = \frac{100}{1+2^{-x}}$$ Ok, i have this expression and my task is to find the inverse. My answer to that is -ln2((100-x)/x). Which is wrong when i test it. Can someone help me with this?
0
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1answer
66 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
0
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1answer
31 views

Number of solutions to an equation

Hello guys I have a simple question to ask. For example I have the equation : $$x^n + x^{n-1} + x^{n-2} + ... + 1 = 0$$ I read somewhere that the number of solutions to an equation is given by the ...
2
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0answers
43 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
0
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0answers
43 views

Equation involving a modulus and variable in an exponent

How would I solve for the first positive non-zero integer value for $x$ in this equation? Equation: $1 \equiv 4^x \pmod{199}$
2
votes
5answers
128 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
0
votes
2answers
43 views

Solve for $m$ in $d^m = n$ [duplicate]

I believe the answer is $m = \lceil \sqrt[d]n \rceil$ or $\lfloor \sqrt[d]n \rfloor$. Can anyone help me?
8
votes
0answers
238 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
1
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2answers
27 views

How to find the x intercepts

$\frac{4}{3} e^{3x} + 2 e^{2x} - 8 e^x$ I have some confusion especially because of the e how can I approach the solution? The solution of the x-intercept is 0.838 Many thanks
12
votes
0answers
245 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
1
vote
1answer
89 views

Equation $e^{\frac{1}{x}} - x =0$

Can someone solve this equations with steps $$e^{\frac{1}{x}} - x =0$$ I dont know how to start. I tried adding logarithms but that doesn't help.
2
votes
0answers
29 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
1
vote
1answer
47 views

Let $f(x) = \exp (x^2 − x + 6)$. Choose Dom(f) so that $f^{−1}$ exists. What is $f^{−1}$ and Dom($f^{−1}$) in your case?

I have already got $$y=\exp(x^2-x+16)$$ $$\ln y = x^2-x+6$$ $$\ln x=y^2-y+6$$ I know for getting inverse function we need to solve for $x$, but what should i do in this case?
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votes
2answers
55 views

Solving exponential equations like $6^{3x}=4^{2x-3}$ using logarithms

I'm trying to solve these using logarithms: $a$) $9^{x+1} = 27^{2x-3}$ $b$) $6^{3x} =4^{2x-3}$ $c$) $210=40(1.5)^x.$ I'm trying to practice logarithms by doing various questions. It's been a ...
0
votes
1answer
42 views

Logarithms and exponential decay

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) $0, 4, 8, 12, 16, 20$ Temperature (celsius) $55, 47, 40, 34, 29, 25$ a) Calculate ...
11
votes
1answer
123 views

Reason for LCM of all numbers from 1 .. n equals roughly $e^n$

I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$. Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a ...
0
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0answers
55 views

Exponential decay of the temperature of coffee

The table describes the cooling of a cup of coffee as it sits on your teacher’s desk in the math office. Time (min) 0 4 8 12 16 20 Temperature (celsius) 55 47 40 34 29 25 a) Calculate a, the ...
0
votes
2answers
62 views

Forming equations for exponential growth/decay questions

Problem Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped ...
1
vote
1answer
61 views

difference between poisson and exponential distributions in the context of client server systems?

I am studying client's request arrival patterns on web and application servers. About web server's request arrival pattern I read that "The request arrival rate on web server follows Poisson ...
0
votes
3answers
113 views

Complex number problem- separating into real and imaginary parts!

Please help with a question that I am working on just now...:) If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I ...
0
votes
1answer
65 views

Prove $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$ [duplicate]

We know that $\lim_{n \to \infty}$ $(1+\frac xn)^n=e^x$. How to prove that $\lim_{n \to \infty}$ $(1+\frac xn-o(\frac 1n))^n=e^x$? Attempt of the proof: Let $\epsilon>0$ $\exists n_0$ such that ...
1
vote
3answers
74 views

Why is $ \overline{e^z} = e^\overline{z} $?

How can you conjugate an entire function? $ \overline{exp(z)} $ I need an equivalent. I thought this is only possible with complex numbers. What is the proof for $ \overline{e^z} = e^\overline{z} $ ...