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

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1
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1answer
19 views

Calculating limits using the definition of number e

I have some examples in Demidovič using this technique and there seems to be no reliable source for them online, so I'll make a small tutorial. Example 1: ...
0
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2answers
42 views

What equation has the form f(x) = n exp(m x)?

I'm a programmer working on a calculation with a curve trend. I'm using OpenOffice Calc (like MS Excel) and it's given me a formula for a graph that I don't understand. I can't find this form ...
8
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6answers
176 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
3
votes
1answer
72 views

Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
1
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1answer
49 views

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $ with $ q \geq 2 $ I'm not sure how to approach this question. I was thinking through in induction with ...
1
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2answers
26 views

Is the exponential function continuous for complex numbers?

Hey this might be a dumb question so here it goes: Is $e^{(x)}$ continuous for $x\in \mathbb{C}$? Specifically this question arose while solving the differential equation in the form of ...
2
votes
1answer
55 views

Find all real solutions for $x$ in $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 .$

Find all real solutions for $x$ in $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 .$ I have found out that the answers were 0,1, and -1. But I used sort of a guess-and check way. ...
0
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0answers
15 views

How to steepen logarithmic function without reducing constant of deceleration

As you can see I have plotted my points in Geogebra and compared them to the function $ y=log_{10}x $ They clearly don't coincide, how would I go about adjusting the function in order to find the ...
0
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1answer
35 views

Integration of $\int_{-2}^{\infty} k^m e^{-a k^2} dk$

How to solve the definite integration as showed in the title. And $m$ is an arbitrary natural number, $a$ is a non-negative number. Many thanks in advance.
3
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4answers
326 views

Exponential Growth

I'm trying to wrap my head around the algebra used to get a solution. The question states: In 2011, the Population of China and India were approximately 1.34 and 1.19 billion people, ...
2
votes
2answers
51 views

Solution of equation $[1+\frac{x}{b}]e^{-x/b}=z$

Can we solve this equation $$\left(1+\frac{x}{b}\right)e^{-x/b}=z$$ We have to determine value of $x$ in term of $z$. Problem occur while calculating the following integral. ...
1
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0answers
21 views

Exponential distribution help

Terry has a part-time job at a call centre. Calls to the call centre occur according to a Poisson process with rate $\lambda$ calls per minute. Terry decide to measure the time that elapses between ...
0
votes
1answer
31 views

How to solve equations containing logarithms and exponentials

Equation 1: $x+e=e^x$ According to Wolfram alpha : Solution of x $\approx$ -2.6 and 1.4 Equation 2: $x-e = \ln(x)$ According to wolfram alpha, Solution for x $\approx$ 0.07 and 4.1 How does ...
0
votes
2answers
68 views

Prove that factorial grows faster than exponential function using limits [duplicate]

How can I prove that the factorial function ($n!$) grows faster than exponential functions (ex: $2^n$) using limits?
0
votes
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
votes
0answers
29 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
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1answer
32 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
47 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
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3answers
73 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
80 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
votes
0answers
30 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
42 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
47 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 ...
6
votes
0answers
44 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
votes
2answers
69 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
65 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
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1answer
23 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
865 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
195 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
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1answer
29 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
votes
1answer
32 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
310 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
votes
0answers
37 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 ...
1
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2answers
82 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, ...
3
votes
2answers
65 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
42 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
votes
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?
1
vote
0answers
40 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
vote
3answers
44 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
votes
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
votes
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
42 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
37 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
41 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
236 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 ...