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

learn more… | top users | synonyms (1)

1
vote
3answers
66 views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
-1
votes
0answers
18 views

How to integrate this function and decide lambda

I want to decide lambda for I also want to integrate the same function. Please help!
0
votes
2answers
74 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [on hold]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
0
votes
0answers
18 views

exponential regression fit error problem

I have the following data and im trying to get an exponential fit. Ive tried a variety of different tools for this, which all seem to give quite a large error margin at the top of the curve. Plotting ...
2
votes
4answers
41 views

Limit of a rational function to the power of x

Ok so I have been trying for days already to find a solution to this all around the web and in math books but to no success. The problem is to evaluate a limit of a function composed by polynomial ...
0
votes
2answers
28 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
1
vote
1answer
24 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
1
vote
3answers
52 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
votes
2answers
22 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
0
votes
1answer
22 views

Proof of simple interest formula

Can someone please prove to me that $I = PRT$, where $P$ is the principal, $R$ is the interest rate, and $T$ is the number of years/time. I have seen $I = P(1+TR) = P+PTR$ which does not equal $PRT$, ...
2
votes
0answers
21 views

Lie group - exponential Diffeomorphism

Let $G$ be a nilpotent, connected simply connected Lie group and $\mathfrak{g}$ its Lie algebra. It is known that the exponential map $\exp$ is a diffeomorphism. Now let $\mathfrak{g}_0$ be a Lie ...
1
vote
3answers
26 views

Programming Help - Solving for e(n)

I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it ...
1
vote
3answers
97 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
0
votes
0answers
16 views

What's the MLE of $\frac1\lambda$ for $f(x)=\frac1\lambda\exp − \frac x\lambda$?

What's the MLE of $\frac{1}{\lambda}$ for $f(x)=\frac{1}{\lambda}\exp \left(− \frac{x}{\lambda}\right)$? Is it equal to the mean of $x$ or the inverse of mean of $x$? Thanks,
-3
votes
1answer
28 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
1
vote
3answers
67 views

Prove that $e^x \gt 0$ for $x \in \mathbb{R}$ [duplicate]

This is a consequence of the exponential rule, but how do I actually prove it to be true?
0
votes
0answers
29 views

Continuous exponential functions

In my book, it makes it appear that any continuous exponential function, such as those regarding money, do not follow the traditional formula of $$\text{growth} = (1+\text{return})^x $$ Rather, it ...
0
votes
0answers
41 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
1
vote
1answer
44 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
1
vote
0answers
19 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
1
vote
1answer
25 views

problem about population growth

At the beginning of the Gold Rush, the population of Coyote Gulch,Arizona was $365$.From then on ,the population would have grown by a factor of $e$ each year,except for the high rate of ...
0
votes
1answer
32 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
0
votes
0answers
66 views

A question about exponential matrices

So here is my question, I would like to prove, If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$ And here is ...
1
vote
2answers
18 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...
0
votes
0answers
18 views

Generic Exponential curve base derivation

Alrighty so I am working on a computer program that forms ADSR envelopes including exponential curves for the attack, decay, and release segments. It uses the following equation for the exponential ...
-1
votes
2answers
21 views

Limit of n * ln(1+x/n)

How can you compute with the most primitive tools that: $$ \lim_{\stackrel{n \to \infty}{n > -x}}n \:\ln (1+\frac{x}{n})=x $$ Using l'hospital verifies this. However we hadn't proofed ...
1
vote
2answers
43 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
0
votes
1answer
29 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
2
votes
2answers
40 views

why is $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$?

I'm having trouble trying proving this fact: $\lim_{t \to \infty}t^m e^{-\alpha t} = 0$ for every $m \in \mathbb{N}$ fixed and $\alpha \in \mathbb{C}$ with $Re(\alpha) \gt 0$ I tried to use ...
0
votes
1answer
39 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
14 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
vote
0answers
29 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
2
votes
5answers
407 views

Which is actually exponential?

I've heard the term "exponential" applied to two sorts of functions: $$n^x\text{, where $n$ is a constant (e.g., $2^x$)}$$ and $$x^2$$ Which is really exponential, and what do I call the other ...
0
votes
1answer
32 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
0
votes
3answers
53 views

Calculate exponential decay

I know this has been asked a few times before, but I'm struggling to apply it to my scenario... I have some known values in a table: $$ \begin{array}{|c|c|} \hline \text{Degrees} & ...
0
votes
1answer
36 views

Length of a very basic exponential curve

I have the beginning points (0,1) and end points (180, 141.732) of a curve. The function I am currently using is f(x) = Ae^kx. However, when deriving the original function, I end up with 0 (from ...
2
votes
0answers
26 views

Showing the exponential and logarithmic functions are unique in satisfying their properties

The question asks to prove that there exists a unique function defined on $\Bbb R$ and satisfying the following conditions: 1) $f(1) = a$ $(a>0, a \neq 0)$ 2) $f(x_1) \cdot f(x_2) = f(x_1 + ...
2
votes
1answer
31 views

Evaluating a limit with two steps - Right/Legal?

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {\left( {{{4{n^2}} \over {(2n + 1)(2n - 1)}}} \right)^{1 - {n^2}}} = \mathop {\lim }\limits_{n \to \infty } {\left( {{1 \over {{{(2n + 1)(2n ...
0
votes
1answer
11 views

Determining the gradient of a decaying function

I have data that can be fitted using an exponentially decaying function: $y = e^{-t/\tau}$ and I want to determine the value of $\tau$. I see that if I make the t-axis logarithmic I get a straight ...
1
vote
1answer
181 views

Problem Solving ( Sequences and series)

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
0
votes
0answers
31 views

Expected waiting time of the process in the queue given that the process is served

Consider the following situation: There is one server with exponential service time with parameter $\lambda$. One process is waiting in the queue. The waiting time is exponential with parameter ...
0
votes
1answer
20 views

solving exponential functions

$(27^{x - 1})(3^x) = 9^{2x-3}$. I apologize if you do not understand the equation. I was unsure on how exactly to represent it correctly. I have gotten to the step in the equation where it is ...
1
vote
1answer
33 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
1
vote
1answer
20 views

Sketching Logs with Quadratic Terms

$\log(x^2+1) = y$ asymptote at $x^2+1 > 0$ and so there is no asymptote $x$ and $y$ intercept at $(0,0)$ How do you know that the function goes both directions, and has a dip in the middle? ...
1
vote
1answer
40 views

Can the variables of $y = A + B \mathrm{e}^{C t}$ be solved analytically given 3 sets of points?

Given the non-linear equation $y = A + B \mathrm{e}^{C t}$ and 3 sets of points: ($y_1$, $t_1$), ($y_2$, $t_2$), ($y_3$, $t_3$), can the variables $A$, $B$, and $C$ be calculated analytically? ...
0
votes
3answers
90 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
1
vote
1answer
36 views

Find the inverse function about a exponential related function

Here is the function:$$y = 4x + {x^m},where{\text{ 0 < m}} \leqslant {\text{1;}}$$ Approximately results is acceptable.
3
votes
4answers
152 views

Absolute value of complex exponential

Can something explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook is telling me.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
8
votes
3answers
119 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
1
vote
1answer
43 views

Product of exponential distributions

Suppose $X_1$ is $\mathrm{Exp}(\lambda_1)$ and $X_2$ is $\mathrm{Exp}(\lambda_2)$. $X_1$ and $X_2$ are independent. Let $Y = \min (X_1, X_2)$ and $Z = \max (X_1, X_2)$ and $W = ZY$ . Compute the ...