Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [exponential-function]

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

1
vote
0answers
15 views

Gradient of the Lie exponential map on SO(n)

I am interested in computing the gradient of $f(e^A)$ when $A$ is a skew-symmetric matrix. If we write $e^A = B$ and we denote the gradient of the function $f$ on the ambient space evaluated at $B$ ...
0
votes
1answer
18 views

Conditions of exponential functions

Okay, is the above statement correct? Because if I put $x=\frac{1}{2}$ then $f(x)$ will have two values. So will that remain a function anymore?
0
votes
0answers
11 views

how to compare two sets of exponential data for precision

I have two sets of exponential data (temperature measurements) of the form T(t)=$-e^{-kt}$. k is a constant that determines the rate of temperature change. 1 temperature measurement was taken every 10 ...
0
votes
0answers
23 views
2
votes
1answer
29 views

Calculating $\sum_{k=1}^\infty 2^{-k}(e^{-k}-e^{-k-1})$ [on hold]

Pretty basic question - I'm just not that experienced with calculating summations and would love help with understanding the steps involved (I computed it with Mathematica but couldn't see the steps ...
1
vote
1answer
27 views

Rewriting this exponential in terms of n

I'm having trouble rewriting this in terms of n. I'm trying to compare this with other asymptotic functions. $$16^{\sqrt{\log_2n}}$$ I have these other ones which I have already ordered: $\log_2 n, (...
3
votes
1answer
46 views

Inequality on the exponential function

By playing around, I seem to have come across the following inequality, valid for all $x$: $$x-(1-e^{-x}) \ge e^{-\frac{2}{x}} x$$ (The constant $2$ is not necessarily the tightest one possible.) Is ...
-6
votes
1answer
30 views

Solve this problem please [on hold]

Solve this in as many ways as possible. If $4^x+5^x=6^x$, find the value of $x$ ?
0
votes
1answer
40 views

How to make squaring method of finding an inverse function to be invertible?

I've been trying to find an inverse of this function $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ These are the approaches First approach using squaring ...
1
vote
0answers
27 views

Connected Lie group 1-dimensional isomorphic to $\mathbb{R}$ or to $S^1$

Let $G$ a connected Lie group of dimension 1. Show that \begin{align} G \cong \mathbb{R} \, \, \, \text{or} \, \, \, G \cong S^1 \end{align} I tried to read and understand the topic Connected, one-...
3
votes
2answers
201 views

Proving inequalities hold when applying exponentials

So I'm set out to prove that for all $a,b\in\mathbb{R}^+$ where $a,b > 0$, and for all $r\in\mathbb{Q}$ where $r > 0$, $$ a < b \quad \text{if and only if} \quad a^r<b^r $$ This seems so ...
0
votes
0answers
28 views

Express in exponential form : 5 bacteria triple every 15 minutes. How many after 3 hours?

How do I explain how to write this in exponential form in the easiest way possible such that a 5th grader can understand? There were five bacteria in a dish at the beginning of an experiment. Every ...
1
vote
2answers
248 views

Exponents and Log

$3^{2x}-2\left(3^x\right)=3$ My solution: $\left(3^x\right)^2-2\cdot \:3^x=3$ Make the substitution $3^x=u$ $\left(u\right)^2-2u=3$ $u^2-2u-3=0$ $u=3,\:u=-1$ No solution for $3^x=-1$ $3^x=3$ ...
1
vote
3answers
49 views

Solve $9^x-2^{x+\frac{1}{2}}=2^{x+\frac{7}{2}}-3^{2x-1}$

$$9^x-2^{x+\frac{1}{2}}=2^{x+\frac{7}{2}}-3^{2x-1}.$$ The equation states solve for $x$. What I first did was put like bases together. $$3^{2x}+3^{2x-1}= 2^{x+\frac{7}{2}}+ 2^{x+\frac{1}{2}}.$$ ...
1
vote
1answer
64 views

What is the mathematical approach of inversing a function resulting in a piece-wise solution?

I've been trying to find the inverse of $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ Here are my steps $$ \begin{align} x & = e^{-\left(\displaystyle \frac{...
-1
votes
2answers
19 views

Even out a numerical curve

I have a set of values that increase / decrease exponentially thus: ...
0
votes
1answer
23 views

What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)

I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and ...
2
votes
1answer
47 views

Prove that $e^{\frac12i\,\text{gd}(\pi t)}\cos^\frac12(\text{gd}(\pi t))=\frac{1+i}{e^{\pi z}+i}e^{\frac{\pi z}2}$

This question was derived from this post about Gamma function. Juan said: $$ \Gamma\left(\frac12+it\right)=\sqrt{\frac{\pi}{\cosh\pi t}}\exp\left\{i\left(2\vartheta(t)+t\log(2\pi)+\arctan\tanh\...
3
votes
1answer
73 views

Not clear in one step in the inequality for the proof LCM$(1,2,\dots,n) < 3^n$

I believe that I am following Hanson's proof up until this point (page 36): $$C(n) < \dfrac{(en)^{k-1+1/(a_{k+1}+1)}w^n}{(a_1)^{(a_1-1)/a_1}(a_2)^{(a_2-1)/a_2}\dots(a_m)^{(a_k-1)/a_k}}$$ where $...
1
vote
1answer
146 views

Prove $F_{n+1} ≤ (\frac74)^n $, where $F_n$ are Fibonacci numbers [duplicate]

Let $F_n$ be the $n$-th Fibonacci number, defined recursively by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n−1} + F_{n−2}$ for $n ≥ 2$. Prove the following by induction (or strong induction): $(a)$ For ...
5
votes
2answers
283 views

Find range of exponent

I have the following function I need to find the range for and I'm not sure if I'm on the right direction. $f(x,y) = e^{-x^2-(y-1)^2}$ $x$ & $y$ are real-numbers. I'm thinking that the range is ...
1
vote
2answers
71 views

$a + b = 2$ implies $a^c + b^c \ge 2$ for any real $c \ge 1$

If $a, b, c$ are positive reals such that $c \ge 1$ and $a + b \ge 2$ then $a^c + b^c \ge 2$. Is there any elementary way to prove it without using calculus and advanced inequalities like Jensen's? ...
2
votes
1answer
77 views

Find $\lim_{x \to 0^{+}} \frac{\pi^{x\ln x} - 1}{x}$ if it exists .

Let $f(x) = \frac{\pi^{x\ln x} - 1}{x}$ . Find $\lim_{x \to 0^{+}}f(x)$ if it exists . My try : $f(x) = \frac{\pi^{x\ln x} - 1}{x} = \frac{e^{x\ln x \ln \pi} - 1}{x}$ . Using $(\forall u\in\mathbb{R}...
-2
votes
0answers
13 views

natural gradient descent

How to calculate/approach Natural Gradient Descent for the general form of Exponential Family distribution? Exponential Family $$p(x|g(\theta)=\exp(\langle g(\theta),T(x)\rangle-F(g(\theta)))$$
0
votes
0answers
19 views

Integrating exponents

From my years of learning in mathematics, I thought no matter how you integrate, the powers of an exponential function cannot change. Could someone point me how to solve for this? Or do they actually ...
2
votes
1answer
44 views

Prove that $e^{\sin x}-\sin e^x\neq0$ $\forall x\in [-1, \pi]$, an equation in the form $f(g(x))=g(f(x))$

As the title say, we need to prove that: $e^{\sin x}-\sin e^x\neq0$ $\forall x\in [-1, \pi]$ Now, I am not sure if this can lead somewhere but we can notice that $e^{\sin x}-\sin e^x=0$ is ...
2
votes
2answers
102 views

Solving $2^x + 3^x = 12$

I need to solve $2^x + 3^x = 12$ for real $x$. I tried the following: $$ 2^x + 3^x = 3\times 2^2 \\ 1+(3/2)^x= 3\times 2^{2-x} $$ But from here on I don't know how to apply logarithms.
1
vote
2answers
39 views

How to compare logs

I have a quick question about simplifying these exponents and then comparing them: $8^{\log_2 n}, 2^{3log_2(log_2n)}$ and $2^{(log_2(n))^2} $ I know the third one evaluates to $n^{log_2(n)}$, but I'...
1
vote
2answers
64 views

General form of the $n^{\text{th}}$ derivative of $x^x$

Can someone help me to confirm this identity that I have established, I really have no idea how I would go about proving that this is true. By the way, this is not a homework assignment, I am just ...
1
vote
2answers
62 views

Given $S_n = \sum \dots$ and $a_n = \sum \dots$ prove that $a_n = S_n + {1\over n\cdot n!}$

I'm trying to solve the following problem: Let: $$ \begin{align} S_n &= 2 + {1\over2!} + {1\over3!} + {1\over 4!} + \dots + {1\over n!} \\ a_n &= 3 - {1\over 1\cdot2\cdot2!} - {1\over 2\...
0
votes
2answers
51 views

$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$ for real $x,y$ with $x\in(0,1)$

For real $x,y$ with $x\in(0,1)$ we have $$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$$ My attempt : Obviously, we have the identity $(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}=(1-x^{3})|1-x\cos y-...
0
votes
1answer
47 views

show that for $a \ge 2, n \ge a, (1 + \frac{1}{(n-a+1)/(a-1)})^{(n-a+1)/(a-1)} < e$

In the proof by Denis Hanson (see Lemma 3), this is presented as obvious if $a \ge 2$ and $n > a$: $$\left(1 + \dfrac{1}{(n-a+1)/(a-1)}\right)^{(n-a+1)/(a-1)} < e$$ where $e$ is Euler's ...
2
votes
4answers
49 views

Find all complex numbers $z$ such that $e^z=1+2i$

I'm not sure how to do this. I would really love it if someone would talk me through it. I know we must somehow solve $$ z=\log1+2i $$ Another idea I had is that we know that $e^{2\pi i}=1$ and $e^{\...
1
vote
2answers
44 views

Find intervals where $f$ is increasing/decreasing $f(x) = e^{-3x} -3e^{-2x} + 1$

I'm doing a problem where I'm asked to find the intervals where $f$ is decreasing or increasing of: $$f(x)=e^{-3x}-3e^{-2x}+1.$$ I've found that the derivative is $f'(x)=-3e^{-3x}+6e^{-2x}$ and ...
8
votes
1answer
128 views

How to solve $32^x - 8 = 2 \cdot 4^x$

I am given the following equation to solve $$32^x - 8 = 2 \cdot 4^x$$ which one can simplyfy to $$2^{5x}-2^3 = 2^{2x+1}$$ where do we go from here? If we had something like $$2^{2x} - 5 \cdot 2^x ...
-1
votes
1answer
28 views

The formula $P = 1,527,000 (1.015)^t$ gives the population $t$ years after 2008. Find the population in 2009 and 2010? [closed]

The formula $P = 1,527,000 (1.015)^t$ gives the population $t$ years after 2008. Find the population in (a) 2009 (b) 2010 What is the answer and a proper solution to this problem? This is all ...
23
votes
15answers
3k views

Simple way of explaining the empty product [duplicate]

I understand that $$3^2=9 \text{ because } 3\times3=9$$ But is it possible to explain in same simple terms how $3^0=1$?
0
votes
1answer
44 views

BCH formula and exponential definition

We denote the flow, of the dynamical system defined on $M$ by $$ \dot x(t) = X(x(t)), $$ by $\exp(tX) ~x(0)$, which is a map from $M$ to $M$. This notation comes from the linear case, where the flow ...
1
vote
4answers
34 views

Regression of Exponential Ramp

Given a set of measured data ( temperature ), I need to estimate parameters of the exponential function which I suppose be the best fit: $y=A+C(1-e^{-t/\tau}$) From an operative point of view, given ...
0
votes
0answers
32 views

Question on the final step of Hanson's proof that $\prod\limits_{p^a \le n}p^a < 3^n$

In Denis Hanson's proof that $\prod\limits_{p^a \le n} < 3^n$, I am confused by the final step. He proceeds to show that: $$C(n) = \frac{n!}{\lfloor n/a_1\rfloor!\lfloor n/a_2\rfloor!\lfloor n/...
1
vote
2answers
37 views

Differentiating an exponential integral

I need to evaluate this expression: $$\frac{d}{dt}\exp\left({-\int_t^T r(\tau) d\tau}\right)$$ I first start with the exponential rule $$\frac{d}{dx}e^{g(x)} = \frac{dg(x)}{dx} e^{g(x)}$$ Now I ...
0
votes
2answers
55 views

Working with the equation: $25^{\log_{10}(x)}=5+4x^{\log_{10}(5)}$

The answer should be $10$, but I don't know how to get it. I've tried many different ways and I still can't get it. I tried this : $$25^{\log_{10}(x)}=5+4x^{\log_{10}(5)}\implies \log_{10}(25^{\...
0
votes
1answer
29 views

How to expend $\log_a(\log_ax)$ for $a\in(0;1) \land x\in(0;1)$?

Here are some logarithm rules : $\log_ay=\frac{\ln y}{\ln a}$ $\log_a(A\cdot B)=\log_aA+\log_aB$ $\frac{1}{\ln a}=\log_ae$ Hence: $$\log_a(\log_ax)=\log_a \left(\frac{1}{\ln a}\cdot \ln x\right) = \...
1
vote
1answer
24 views

Exponential decay with a discrete step

I'm quite sure it's a dumb question, but I'm struggling with it. I have a modelization in which the state/value of a variable is updated every $2u$. This is the discrete step I'm using. The variable ...
3
votes
1answer
61 views

How to solve for a in the below equation:

I have the following equation $$\int_0^1 \frac{a^x -1}{a-1} dx = r,$$ where the integral evaluates as: $$\frac{1}{1-a} + \frac{1}{\log(a)} = r.$$ I would like to solve for a but this is proving ...
1
vote
1answer
21 views

Getting function from four points

I'm facing this problem I can't solve myself. I've got four points on a cartesian place, and I would like to find the function that equates them. Coords are: ...
0
votes
0answers
19 views

Approximating exponential of differential operator acting on the exponential of a function

I am looking for an approximation of the following expression $$z(x)=\frac{\mathbf{e}^{-\nabla^2} \mathbf{e}^{f(x)}}{\mathbf{e}^{f(x)}}.$$ It is known than the exponential operator can be expanded ...
5
votes
1answer
100 views

Prove: for all $x \in (0, 1], 2^x+2^{\frac{1}{x}} \leqslant 2^{x+\frac{1}{x}}$

This problem is from my math teacher.I tried using Calculus, the derivative function is like a black hole.Then I graphed it by Mathematica. As the following picture shows, I was strongly astonished. ...
3
votes
1answer
89 views

Evaluating $\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$

Is there a closed form for the following integral? $$\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$$ Nothing of this form seems to appear in Gradshteyn and Ryzhik.
0
votes
0answers
35 views

Inequality - Logarithm of the Sum of Exponentials

Is it true that for $0<\alpha<1$ and $x_1,\dots,x_n,y_1,\dots,y_n\in\mathbb{R}$ we have the inequality: $$ \log \sum_{i=1}^n e^{\alpha x_i+(1-\alpha) y_i} \leq \alpha\, \log \sum_{i=1}^n e^{x_i}...