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

learn more… | top users | synonyms (1)

1
vote
0answers
18 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
37 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
42 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 ...
-1
votes
0answers
16 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
0
votes
1answer
30 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
votes
0answers
36 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
votes
0answers
19 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
124 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
32 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?
5
votes
0answers
103 views
+100

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
vote
2answers
22 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
-4
votes
1answer
32 views

Exponential Math Functions [on hold]

Iodine — 131 is a radioactive substance used in nuclear medicine. Suppose a patient was given a dose of 6mL. The half-life of Iodine-131 is 8 days. Determine the amount of iodine-131 in the patient ...
6
votes
0answers
84 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} ...
2
votes
1answer
79 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
27 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 ...
0
votes
0answers
25 views

math in medicine

Use the following parameter values for the model of the periodic arterial pulse that we considered in class: Rs = 17:5 mmHg/(liter/min), Csa = 0:00175 liters/mmHg, V0=0.07 liters, T = (1/80) min. ...
1
vote
1answer
40 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?
-2
votes
2answers
43 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
21 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 ...
7
votes
1answer
59 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
votes
0answers
9 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 ...
-1
votes
2answers
32 views

Forming equations for exponential growth/decay questions [closed]

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 up when ...
1
vote
1answer
27 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
54 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
58 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 ...
0
votes
3answers
67 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} $ ...
1
vote
1answer
44 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
2
votes
2answers
44 views

yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
-1
votes
0answers
13 views

Integration Bessel Function & Exponent Function & Trigonometric Function [closed]

I have an integral with the form I[r]=∫(arExp[-r]-brSin[k(r-d)]Exp[-r])Besse1J[0,kr]dr where Besse1J[0,kr] is the modified bessel function of the first kind and a, b, k, d are constants. The ...
0
votes
1answer
71 views

Solve exponential equation $6\times3^{2x}-13\times 6^x +6\times 2^{2x}=0$

I have tried solving the following equation by using exponential properties and logarithms, but can not find some link between all of the terms: $$6\times3^{2x}-13 \times6^x +6\times 2^{2x}=0$$ ...
1
vote
1answer
22 views

Exponential Growth and Decay / compound interest

This is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?" ...
1
vote
2answers
57 views

Integral $\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$

Is there any closed form expression for the definite integral $$\int_0^b \frac{1-\exp\{-x\}}{x}\text{d}x\qquad 0<b<\infty$$ as I could not find one in Gradshteyn and Ryzhik Table of Integrals?
0
votes
1answer
31 views

Point of intersection between two exponentials with a constant term

Is there any way to solve algebraically for $x$: $a^x - b^x = C$ If not, is there a commonly used function that can be used to represent its solution? e.g., the Lambert W function for $a^x - bx = C$ ...
4
votes
1answer
21 views

Exponential Growth and Decay : $y = a (1+r)^t$

I know this is a really basic question for this website, but I can't find it anywhere else. This is the question: "If you deposit $\$3,750$ in an account that pays $6\%$ annual interest compounded ...
1
vote
1answer
25 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
1
vote
7answers
151 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
2
votes
0answers
71 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
4
votes
1answer
71 views

Solve the integral [closed]

Can anyone solve these two integrals . $$ \int_{0}^{ \infty } \frac{x^2 e^{-x^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$ and $$ \int_{0}^{ \infty } \frac{e^{-(\ln x - \mu )^2/2 \sigma ...
-1
votes
3answers
90 views

Which is greater $e^{\pi}$ or $\pi^e$? [duplicate]

Recently I asked a question on Maths SE Proof that at most one of $e\pi$ and $e+\pi$ can be rational after solving this one one I was thinking whether $e^\pi$ is greater or $\pi^e$ ? On calculating ...
1
vote
1answer
46 views

Exponential function given two points

I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that ...
0
votes
1answer
50 views

Exponential Equation $4\cdot7^{x+2}=9^{2x-3}$

Let $4\cdot7^{x+2}=9^{2x-3}.$ I do not know how to solve for $x$. Progress Took logarithms, got $$4(x+2\log7)=(2x-3)\log9$$ $$(x+2)\log7=[(2x-3)\log9]/4$$
0
votes
1answer
35 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}} \exp\left(-\frac{m}{\gamma} x \right) dx. $$ I tried to look into different books such as Gradshteyn and Prudnikov ...
0
votes
3answers
60 views

Adding complex exponentials

Can somebody please explain $$e^{-\frac{3}{4}\pi i}+e^{-\frac{9}{4}\pi i}+e^{-\frac{15}{4}\pi i}+e^{-\frac{21}{4}\pi i}=0$$ WolframAlpha Computation.
0
votes
2answers
46 views

what to do with: logarithmic, trigonometric and exponential inequalities with variable outside

After encountering this inequality: $$ e^{x/2}=2x+1 $$ that leads me to: $$ x=2\ln(2x+1) $$ I realized that I don't know how to solve it. But this lack of knowledge expands also to $\cos(x)=x$ or ...
0
votes
0answers
24 views

Exponential convergence of controlled variables

I am reading a paper and I don't understand why, after some math they say that the controlled variables $$ \dot{\psi}_{13} $$ and $$ \dot{\psi}_{23} $$ converge exponentially. This is the paragraph ...
0
votes
0answers
53 views

Study $f_{\lambda}(x) = \lambda e^x + x^2 + 2x +2$ for any $\lambda \in \mathbb{R}$

This time I have the following questions: Consider $$f_\lambda: x \longmapsto \lambda\exp(x)+x^2 +2x +2$$ for any real $\lambda.$ 1) Compute $f'_\lambda$ (the derivative of $f_\lambda$). Show ...
0
votes
2answers
60 views

Calculating $ \lim_{n\to \infty} (1+\sin({1}/{n}))^{n}$ without L'Hopital or series expansions [duplicate]

I am trying to calculate the following limit, without using the L'Hopital rule or series expansions: lim (1+sin(1/n))^(n), n->infinity I now that it is the ...
1
vote
0answers
22 views

Prove convergence of $(1-\frac xk)^k$ as $k\to\infty$ using arithmetic-geometric mean

Define $f(x):=x^{t-1}e^{-x}$. For $k=1,2,\dots$ let $$f_k(x)=\begin{cases}x^{t-1}\left(1-\frac xk\right)^k & 0<x<k\\0&k\le x\le \infty\end{cases}$$ Show that $f_k(x)\to f(x)$ and ...
1
vote
1answer
34 views

Rewrite formula using exponential generating functions

In the equation below I want to extract $b_k$. $$\frac{a_n}{n!} = \sum_{k=0}^{n}\frac{b_k}{k!(n-k)!}$$ For all other exercises in the book I had to use $e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$ or ...
0
votes
1answer
28 views
+50

Transformation Ricker equation

The classical Ricker equation for modelling density-dependent population growth is: $N_{t+1} = a_t * e^{r * \left(1-\frac{N_t}{k}\right)}$ where $N_t$ is the initial number of individuals (starting ...