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

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14
votes
2answers
80 views

$\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
0
votes
4answers
36 views

Law of Natural Logarithms

This is an old example and since I've free time, I am working on it. $$B(t)= \frac{12}{1+e^{-0.6(t-6)}} $$ If we set $$10= \frac{12}{1+e^{-0.6(t-6)}}$$ $$ \ln 10 = \ln \...
1
vote
1answer
30 views

How do I know if this equation can be solved symbolically?

Can these equations be solved symbolically for $x$? $$ \begin{align} x &= \frac{p - p_m(x)}{p_m(x) - p_m(x)^2} \\ \\ p_m(x) &= \frac{e^x}{e^x + e^y} \\ \end{align} $$ If not ...
1
vote
4answers
94 views

If $n$ is a positive integer, then $(-2^n)^{-2} + (2^{-n})^2 = 2^{-2n+1}$

I'm not sure why $$(-2^n)^{-2} + (2^{-n})^2=2^{-2n+1}$$ I have been going over this equation for a while now, noticing, and have successfully got quite far in the equation, finding that $$ (-2^n)^{-...
-4
votes
3answers
45 views

Exponential equation problem with no solution? [on hold]

I have trouble solving this: $$3^x+3^{2-x}=8$$ I have tried substituting $3^x=z$ but that doesn't seem to help much.
3
votes
2answers
151 views

$\pi$ and $e$ as coded trajectories

Question about the number $\pi$ and $e$ and their unpredictability. We know that $\pi=3.141592653589793238462643383279502884...$ Suppose that we are in the origin of the plane i.e. at the point $(0,0)...
1
vote
1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
0
votes
1answer
48 views

What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$ e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k $$ Is ...
2
votes
1answer
44 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
1
vote
3answers
102 views

Why does $a\cdot r^{-1}$ equate to $\frac {a}{r} = 1$?

Why is $a\cdot r^{-1}=1$ equivalent to $\frac {a}{r} = 1$? I am trying to write exponential functions from graphs; two points were given: $(-1,1)$ & $(-2,5)$. I am trying to find an equation ...
4
votes
3answers
143 views

Solve 3 exponential equations $z^x=x$, $z^y=y$, $y^y=x$ to get $x$, $y$, $z$.

The main question is : $z^x=x$, $z^y=y$, $y^y=x$ Find $z$, $y$, $x$. My method : I first attempted to get two equation for the unknowns $x$ and $y$. We can happily write : $z=x^{1/x}$ and $z=y^{...
0
votes
3answers
37 views

The value of an investment in Canada Savings Bonds is modelled by $A(t) = A_0 e^{0.0255t}$… Rest of question below.

The value of an investment in Canada Savings Bonds is modeled by $$A(t) = A_0 e^{0.0255t}$$, where A is the amount the investment is worth after $t$ years, and $A_0$ is the initial amount invested. At ...
0
votes
0answers
18 views

Exponential curve fit with MATLAB's fit function does not deliver good fit

I am trying to use MATLAB's fit function to fit a curve through a data set which obviously shows an exponential decay. These are the commands I use: ...
-1
votes
0answers
34 views

Exponential equations in one variable for the reals [closed]

My father approached me yesterday and asked me if I could solve $$4^{x}+5^{x}=6^{x}$$ I countered by asking him over what set. He told me $R_{>0}$ So by using the intermediate value theorem it's ...
0
votes
0answers
22 views

Logarithm's inequality correctness

It is well known that for , the following holds: Now, given a set of n points, P, is the following term right for every and for every : If so, how can i prove that the term exists? And if it ...
0
votes
0answers
16 views

finding the minimum and maximum values for $(q+r)$.

given positive integer $p$, $q$, and $r$ with $p=3^q\cdot2^r$ and $100<p<1000$. find the difference between maximum and minimum values for $(q+r)$. I did find the answers by hit and trial ...
7
votes
3answers
489 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\...
0
votes
2answers
94 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $\log()$ the value $e$... and note that so far ...
1
vote
1answer
21 views

Limit of a function with exponential function and two parameters tending to infinity

I need some help with calculation of limits. I have a function $n(e^{it/\sqrt{m+n}}-1) + m(e^{-it/\sqrt{m+n}}-1) + \frac{m-n}{\sqrt{m+n}}it$ The solution says this converges to $-\frac{1}{2}t^2 \...
0
votes
3answers
28 views

How to divide exponents with different base numbers

Could not find a calculator online that could handle my large number. Could some help me with the solution for this very large number, I've forgotten how to divide exponentials with different bases. $...
1
vote
1answer
29 views

Which function is approximately equivalent to $C(t) = 10(1.029)^{24t}$?

I am looking over at some math questions and I encountered this problem: The growth of a certain organism can be modeled by $$C(t) = 10(1.029)^{24t},$$ where $C(t)$ is the total number of ...
1
vote
4answers
165 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
1
vote
0answers
23 views

When is $\frac{2 n f(n)}{n !}$ in the order of some fixed power of $n$?

I would like to know when $\frac{2 n f(n)}{n !}$ is $O (n^b)$ where $b$ is a constant. Here, $n$ is a positive integer. My attempt: $$ \frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{...
-3
votes
2answers
38 views

Determine the equation of the tangent to y=3(2^x) at x= 3 [closed]

I am not quite sure how to go about this question. If you could help that would be great.
2
votes
1answer
57 views

Laurent series with $e^z$

I'm trying to find the Laurent series Expansion for $$ f(x) = \frac{e^z-(z-1)}{z-1} $$ on the annulus $0<|z|<\infty$. I'm aware that I am supposed to use substitution of known series. I am ...
1
vote
2answers
33 views

Solution of composition of function

In a book I saw a question along with solution The question is Let f,g,h be function from R to R , then show that (f+g)oh = (foh).(goh) But when I saw the solution i got confused , they have ...
14
votes
5answers
352 views

$e^{\left(\pi^{(e^\pi)}\right)}\;$ or $\;\pi^{\left(e^{(\pi^e)}\right)}$. Which one is greater than the other?

$\newcommand{\bigxl}[1]{\mathopen{\displaystyle#1}} \newcommand{\bigxr}[1]{\mathclose{\displaystyle#1}} $ $$\large e^{\bigxl(\pi^{(e^\pi)}\bigxr)}\quad\text{or}\quad\pi^{\bigxl(e^{(\pi^e)}\bigxr)}$$ ...
4
votes
2answers
75 views

For which classes of matrix can the matrix exponential be easily computed?

We have diagonal matrices $A = \mbox{diag} (\lambda_1, \ldots, \lambda_n)$ for which matrix exponential has simple form $e^A = \mbox{diag} (e^{\lambda_1}, \ldots, e^{\lambda_n})$, and it can be ...
0
votes
0answers
18 views

Generalized exponential and logarithmic functions

The $q$-exponential and $q$-logarithmic functions are defined as in here. Does any one know whether this definition can be extended to $q=\infty$?
1
vote
2answers
668 views

Change of variables for triple integrals

Question: Consider the one-to-one transformation $(u; v;w) \to (x; y; z)$ defined by the equations $u = x + y + z; uv = y + z; uvw = z;$ which maps the unit cube $U$ defined by $0 \lt u \lt 1, 0 < ...
2
votes
1answer
14 views

Proof that the sum of two independent exponential random variables is gamma with $\alpha=2$

I'm trying to prove that the sum of two exponential random variables is gamma. This proof is straightforward using the uniqueness of moment generating functions however I'm asked to find the density ...
0
votes
0answers
21 views

Terminology for a process with subcritical, critical, and supercritical cases?

I've noticed that, in a number of domains in pure and applied mathematics, there are processes or structures involving exponential growth or decay where the process splits into three cases: a ...
2
votes
6answers
120 views

General solution for $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$?

Start with $$\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$$ then $$\frac{1}{\mathrm{d} x} \, \mathrm{d} \left(\frac{\mathrm{d} y }{\mathrm{d} x}\right) = y$$ $$\frac{\mathrm{d} y}{\mathrm{d} x} \, \...
1
vote
1answer
41 views

Limit of a Function with help of Euler's Formula

I've been trying to get the limit of a function, but I don't know how. The function is $\displaystyle{10^{n}\left(1 - \mathrm{e}^{\mathrm{i}t/10^{\,n}}\,\right)}$ and the solution says this ...
0
votes
2answers
53 views
0
votes
4answers
174 views

Evaluate $\int_{0}^{\infty} (-1)^{\lfloor x\rfloor}\cdot e^{-x} dx $ [closed]

I'm having trouble integrating the following: $$\int_{0}^{\infty} (-1)^{\lfloor x \rfloor}\cdot e^{-x} \, \mathrm{d}x $$ where $\lfloor x \rfloor$ denotes the floor of $x$. Can you help please?
1
vote
1answer
38 views

Exponential Probability of Sum [closed]

Given : $X_1, X_2, X_3, \ldots,X_{10}\ \sim\ \,\mathrm{e}^{\lambda}$ $X_1, X_2, X_3,\ldots,X_{10}\,\,\, \mbox{are independent variables}$ $\lambda > 0.5$ Calculate the following: $$ P\left(\...
1
vote
1answer
23 views

$\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$?

Assuming that $f$ is bounded, continuous, and non-negative, is it true that $$\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$$ I have a hard time proving this ...
2
votes
2answers
94 views

Proof of $\pi^e$ and $e^\pi$ Being Irrational

By contradiction, if $\pi^e$ were rational, then we could write $\pi^e=\frac{a}{b}$ where $a,b\in\mathbb{I}^+$ and $b\neq0$. So: $$\begin{align} \\ \pi^e&=\frac{a}{b} \\ e\ln(\pi)&=\ln(a)-\...
6
votes
1answer
245 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how can ...
0
votes
0answers
25 views

Exponential equation involving different bases.

Alright, I know you can get the solution $ x=2 $ just by giving the equation a glance, but I am asking for a rigorous proof here. All mathematical tools are encouraged (no syllabus limitation). $$ 5^{\...
0
votes
2answers
2k views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
2
votes
0answers
30 views

Analytical integration of product of exponential functions

I am trying to obtain an analytical formula for the following integral. My first question is whether it is possible to obtain an analytical formula without the use of transcendental functions. My ...
0
votes
2answers
127 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ (e^{i\...
1
vote
2answers
44 views

What is $e^{A}$ where A is an anti-diagonal matrix

I am trying to get a closed form for the matrix produced by the following operation: $$e^A$$ where $A$ is an anti diagonal matrix, say, of size $2\times 2$: $$A=\begin{pmatrix} 0 &b \\ c &0 \...
0
votes
3answers
40 views

Subtracting powers with variable in exponent

I am having some troubles with a question that subtracts powers. Solve for unknown: $$3^{x+4} - 5(3^x) = 684$$ I have a hunch that I should apply factorization somehow. Do I multiply 5 and 3 to ...
0
votes
1answer
28 views

Exponent Laws and the Ceiling Function

Suppose I have $f(x) = 5^{\lceil \frac x 3 \rceil}$, where $x \in \Bbb N$. If I were to simplify $f(x+4)$, can I do the following: $f(x+4) = 5 ^{\lceil \frac {x+4} 3 \rceil} = 5^{\lceil {\frac x 3} \...
0
votes
2answers
18 views

Calculating Children in a Hierarchy

I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes. 5 wide ...
2
votes
1answer
28 views

Convergence of exponential of monotonic function?

Let $f:{\mathbb R}_+\rightarrow{\mathbb R}_+$ be an increasing continuous function. We now that $$\lim_{r\rightarrow \infty} \left(1+\frac{f(x)}{r}\right)^{r} =e^{f(x)}.$$ Then $$\lim_{r\rightarrow \...