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

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0
votes
1answer
24 views

If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.

The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$. My method : I first began by substituting $x^2-5x+5$ as ...
15
votes
3answers
124 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 ...
0
votes
4answers
37 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 \...
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}=\...
-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.
1
vote
4answers
95 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)^{-...
0
votes
1answer
50 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 ...
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 ...
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 ...
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 ...
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^{...
1
vote
1answer
35 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 ...
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
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 ...
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 ...
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
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.
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
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 ...
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 ...
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}} $ $$\large e^{\bigxl(\pi^{(e^\pi)}\bigxr)}\quad\text{or}\quad\pi^{\bigxl(e^{(\pi^e)}\bigxr)}$$ ...
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 $q=\infty$?
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 ...
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 ...
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
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 ...
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(\...
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)-\...
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 ...
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^{\...
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 ...
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
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 \...
0
votes
0answers
12 views

Is there any relationship between modified Bessel function of the first kind and exponential function?

Recently I read a book and in that book, I have a question about the relationship between modified Bessel function of the first kind and exponential function. To be more specific, The original ...
1
vote
1answer
22 views

Average Percent Rate of Change

Excuse the png equations, still a MathJax newbie. I am analyzing data I have computed: Alcohol content and Caffeine content retention after a duration of 8 hours for each. I had gotten the data in ...
1
vote
2answers
33 views

I am skeptical of my results (percent rate of change)

Excuse the png equations, still a MathJax newbie. I am analyzing data I have computed: Alcohol content and Caffeine content retention after a duration of 8 hours for each. I had gotten the data in ...
0
votes
3answers
20 views

Reformulate a term

How did we got this? We had to find $T$. From: $$\frac{R}{R_1}= e^{b(\frac{1}{T}-\frac{1}{T_0})}$$ This: $$T= \frac{b T_0}{T_0\ln R-T_0\ln R_0+b}$$
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?
0
votes
2answers
53 views

How to solve this equation ? - Exponential equation

How do i solve that equation algebraically ? Thanks!
4
votes
1answer
162 views

Find the roots of $e^x+e^{1/x} + a = 0$

Find the roots of this equation $e^x + e^{1/x} + a = 0$ where $a \in \Bbb R$ Is there any nice formula for this type of equation?
0
votes
2answers
47 views

Limit of $\sum\limits_{i=1}^\infty \frac{1}{a^i}$ as x -> infinity

I observed something while working a bit about series. I found out that the limit of : $$\sum\limits_{i=1}^x \frac{1}{a^i}$$ as x approches infinity seems to be equal to $\frac{1}{a-1}$. If this ...
0
votes
1answer
35 views

Algebra question about simplying a constant from exponential [closed]

i've a question, i'm doing an exercise of differential equations, but my result is wrong due to a step that compared with wolfram alpha i don't understood. You can check the screenshot, how the $C$ ...
5
votes
3answers
114 views

Integrate $e^{-x^4+x^2}$

I am looking for pointers on how one might approach the following definite integral: $$ \int_{-\infty}^\infty e^{-x^4 + x^2}\, dx$$ Or more generally: $$ \int_{-\infty}^\infty e^{-x^4 + \alpha x^2}\...
1
vote
1answer
25 views

If $x(z)$ and $y(z)$ are analytic with $x(0) = 0 = y(0)$ then $x(z)^{y(z)} \to 0$ as $z \to 0$

I'm a programmer and my math is a little rusty, but usually sufficient for my needs. However, I came across the following statement in the exp(3) manual page of ...
2
votes
2answers
73 views

Simplification of $(e^{2x}-1)/(e^x-1)$

Why is $$\frac{e^{2x}-1}{e^x-1}$$ equal to $e^x+1$ ?