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

learn more… | top users | synonyms (1)

0
votes
0answers
6 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
3
votes
0answers
36 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
1
vote
2answers
19 views

Special vs. General Case in Basic Algebraic Notation

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. : To compute the derivative of a function of the form eg(x), write eg(x) as a composite eg(x) = f(g(x) ), where ...
0
votes
1answer
38 views

Find exponential decay equation for tiger population model

I've forgotten how to do it first it starts.. In 1900, there were 100,000 wild tigers worldwide; in 2010 the number was 3200. (a) Assuming that the tiger population has decreased exponentially, find ...
0
votes
1answer
57 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_{0}^{\infty}\frac{e^{- x f}}{m+x}\gamma\left(a,h x\right) dx$$ where $f,m, and h \in reals$ f,m, and h are $>$ 0 $a \in integers$, a=1,2,3,... ...
5
votes
1answer
74 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
2
votes
4answers
145 views

The definition of e by limits of $(1+1/n)^n$ through series expansion

I think the problem I have is due to not being knowledgeable about limits. If I use binomial expansion to expand $(1+1/n)^n$ to $1 + \frac{n!}{(n-k)!k!}*(1/n)^k + ...$, I can imagine replacing $n$ ...
3
votes
0answers
40 views

Why is the base of an exponential function limited to the set of real numbers greater than zero?

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman. An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$. Why is $b$ ...
0
votes
2answers
54 views

What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ ...
0
votes
0answers
32 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
0
votes
2answers
39 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
2
votes
2answers
58 views

Evaluating exponential integral

I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ $N$ is a ...
6
votes
3answers
114 views

Is $x^x$ a polynomial, an exponential or both?

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function. Is there ...
0
votes
2answers
35 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
0
votes
3answers
24 views

How do I rewrite a logarithm in exponential form, so as to plot it? $f(x) = 2\log x$

How do I write $f(x)=2\log x$ in exponential form? Is $2(10)^y=x$ correct?
-1
votes
1answer
28 views

Exponential function, domain of definition

I have the function $\displaystyle f(x,y)=x^2e^{-x^2-y^2}$ with the domain of definition = $\{(x,y) \mid x^2+y^2=2\}$ The task is to decide $f$'s maximum and minimum value and the range. How do I get ...
0
votes
1answer
21 views

Integrals with an imaginary linear term in the argument of the exponent

in this entry on Wikipedia stays $$ ...
2
votes
0answers
29 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
1
vote
2answers
93 views

Exponential function, multivariable calculus

I got the function $f(x,y)=e^{-x^2-y^2}$ with the domain of definition $x^2+y^2\leq25$. The task is to decide the biggest and lowest value. How do I get there?
2
votes
1answer
65 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
3
votes
1answer
55 views

Solving exponential equation $e^{x^2+4x-7}(6x^2+12x+3)=0$

How would you find $x$ in: $e^{x^2+4x-7}(6x^2+12x+3)=0$ I don't know where to begin. Can you do the following? $e^{x^2+4x-7}=1/(6x^2+12x+3)$ and then find $ln$ for both sides?
1
vote
1answer
44 views

Find real-valued sequences $x(n)$ for which $c^{x(n)} = o(1/n )$

For which $x=x(n)$ does it hold that $$c^x = o\left(\frac{1}{n}\right)$$ where $c\in(0,1)$ is a constant. So clearly, for $x=n$, this is true. But for which $x =o(n)$ does this hold? I thought ...
6
votes
1answer
353 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
0
votes
1answer
27 views

How to scale a equation e.g. by log

I'm currently trying to scale an equation since the numbers I have to calculate with are pretty large and Matlab outputs Infinity (Inf). However, the question here is more about the mathematics behind ...
0
votes
2answers
40 views

Construction of complex numbers and exponent rules for them

I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers. $1$. Is it enough to define it as $a+bi$, where a,b are ...
-1
votes
4answers
44 views

Exponential function (t)

I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.
0
votes
1answer
54 views

How can we know that x^x is an exponential function or not without drawing the graphic?

In general, exponential function is defined as a.b^x, where a=coefficient, and b= base. I only knew that the function is exponential function or not, just by drawing the graphic. But, how can we ...
4
votes
1answer
55 views

Where do I make mistake on this derivative containing e^x^2

My brother is preparing for the university and asked me the following multiple choice question. $$\frac{d}{dx}(x^3 * e^{x^2})$$ a) $e^{x^2}*x^2*(1+2x)$ b) $e^{x^2}*x^2*(3+2x)$ c) ...
0
votes
1answer
13 views

Explicit and Recursive Exponential Growth

The population of a certain organism triples every hour. Write a function that models this growth. By what factor does the population grow in one-half hour? I'm unsure of how I should approach the ...
2
votes
4answers
66 views

How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r ...
1
vote
1answer
36 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
0
votes
0answers
22 views

Log(x) or Exp(x) with limits 0,0 and 1,1

trying to find an equation or function that displays an accelerated progression. It may be an Exp or Exp-like function. Input is X, ranging from 0 to 1. Output is Y ranging from 0 to 1. Note Y must ...
4
votes
4answers
70 views

Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
1
vote
2answers
31 views

What kind of mathematical operation is used to repeatedly increase a number by a certain percentage?

I am sure that this is an easy question to answer for most of you. I need to take a number, let's say $10$, and then increase it by a percentage, let's do $25\%$. $10 \times 1.25 = 12.5$ Easy ...
-1
votes
0answers
15 views

Consistency of the MLE of the exponential distirbution

Is the Maximum Likelihood Estimate of the exponential distribution consistent? I have deduced that it is biased, but am unsure how to show that is consistent. An informal argument is fine, a proof is ...
2
votes
1answer
75 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
2
votes
2answers
61 views

Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not ...
0
votes
0answers
14 views

Explicitly relating two functions containing exponential terms

I have two functions related to the distribution of administered drugs in the body: $$\begin{align}c_1(t) &= a_1\exp(-k_{11}t) - b_1\exp(-k_{21}t)\\ c_2(t) &= a_2\exp(-k_{12}t) - ...
1
vote
5answers
78 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
1
vote
1answer
52 views

Why can we first take the limit that goes to e?

For example \begin{equation} \begin{aligned} \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} \right)^{n} &= \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} ...
3
votes
6answers
95 views

integral of $\frac{1}{(1+e^{-x})}$

I make the substitution $u=1+e^{-x}$ which gives $-\dfrac{e^x}{u}\ du$. Integrating gives me $$-e^x\ln(1+e^{-x}) + C,$$ but the answer is $\ln(e^x +1) + C$. What am I doing wrong?
5
votes
2answers
145 views

$\frac{db^x}{dx}$ without $e$

For no other reason other than interest, I'm trying to find the general derivative of $b^x$ without using a definition of $e$ from a different context. I feel like, chronologically in history, this ...
1
vote
2answers
34 views

Integral of a sum of complex exponentials

Let $$\hat{\varphi_n}(t)=\frac{1}{n}\sum_{j=1}^n{exp(i{t}Y_j)}\quad(t\in\mathbb{R})$$ denote the empirical characteristic function of the residuals $Y_j\,=\,S_n^{-\frac{1}{2}}(X_j-\bar{X}_n),\quad ...
2
votes
2answers
47 views

Isolate x in this equation

I'd greatly appreciate it if someone could please isolate "x" by manipulating the following equation: $$(2^xR)+x=(x-1)p$$
1
vote
4answers
89 views

How many solutions $k>1$ does the equation $\exp ((k-1)/( k+1))=\sqrt{k}$ have?

I have the following equation: $e^{\frac{k-1}{k+1}}=\sqrt{k}$. The question is: how many solutions does it have? ($e$ is Euler's constant and k is a positive real number >1).
5
votes
4answers
168 views

Integral of $\int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$ \int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x $$ $$ 0< y_1 < 1 $$ $$ 0< y_2 < 1 $$ Or is there an ...
2
votes
4answers
135 views

Solving the power equation $A^X = \frac{(1+X)}{(1-X)}$

I want to solve the following power equation (get $X$ value): $$A^X = \frac{(1+X)}{(1-X)},$$ where $X\neq 0$, $A\in {\mathbb R}$ (a real number) $$A \geq 0 , \quad A \leq 1$$ I think $X$ should be ...
4
votes
3answers
285 views

Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\,-\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x $$ Or must I settle with an approximation? Edit: Actual form of integral ...
2
votes
2answers
66 views

Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
1
vote
2answers
48 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...