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

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1
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5answers
91 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
54 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
102 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
149 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
41 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
53 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
96 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).
7
votes
4answers
197 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
141 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
4answers
339 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 ...
1
vote
2answers
65 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 ...
1
vote
1answer
79 views

independent Exponential distribution P(X > Y + 1)

$X$ and $Y$ are independent exponentially distributed random variables with parameters $a$ and $b$. Calculate $P(X > Y + 1)$. I have let $X-Y=Z$ and Then $P(Z>z)=1-P(Z\leq z)$ $1 - P(X-Y\leq ...
1
vote
4answers
55 views

Exponent calculation

How to calculate the decimal powers of any number? (without using log ) Example: $$10^{0.3010} \approx 2$$ I have asked to my maths teacher and many such persons and no one knows the answer. The ...
-2
votes
2answers
64 views

Simple equation $2^x = 16$ [closed]

Solve the following equation: $$2^x = 16$$ What is $x$? For $x = 4$, how do the $16$ and $2$ relate?
1
vote
1answer
94 views

Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?

Let us assume that $f(n)=2^{n+1}$, $g(n)=2^n$ be two functions. Now, use limit to find $O(f(n))$: $\lim_{n\to\infty} \dfrac{2^{n+1}}{2^n}=2$. This is not equal to infinity, so the limit exists, hence ...
1
vote
0answers
42 views

Approximations for finite n in limit-based definition of the exponential function

The exponential function can be defined via: $$ e^x = \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} = \lim_{n \rightarrow \infty} g(x; n) $$ In my problem, I actually have the right ...
6
votes
1answer
112 views

Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.

We know that if two matrix $A$ and $B$ commutes then $\exp(A+B)=\exp(A)\exp(B)$. I am trying to find two matrix that does not commute but $\exp(A+B)=\exp(A)\exp(B)$ is true for them. Can anybody give ...
1
vote
2answers
73 views

Identifying the exponential function $f(x)=e^x$ from its functional equation

Prove that if $f(x+y)=f(x)f(y)$ for all $x,y$ and $f(x)=1+xg(x)$ where $\lim_{x\to 0}g(x)=1$, then: a) $\exists f'(x)$ $\forall x$ b) $f(x)=e^x$ I would really appreciate your help.
5
votes
1answer
115 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n\left(z\right)=\left(1+\frac{z}{n}\right)^n$ for $\:n\in \mathbb{R^+}$. Then ...
2
votes
1answer
23 views

normal equations of $ y(t) = \gamma e^{\lambda t} $ for minimizing the error

Let $ y(t) = \gamma e^{\lambda t} $ and we have the points $(0,2)\ (1,0.7)\ (3, 0.3)$. The task is to get the parameter so that error is minimal. So we need to get the matrix for the normal ...
2
votes
3answers
40 views

inverse of quadratic log functions

Can a Log function with a quadratic have an inverse function? The specific question is to find the inverse of $$f(x) = \log_2(x^2-3x-4)$$ The function already fails the horizontal line test, but ...
5
votes
4answers
287 views

Another method for limit of $[e-(1+x)^{1/x}]/x$ as $x$ approaches zero

I have solved this limit: $\lim_{x \rightarrow 0} \frac{e-(1+x)^{\frac{1}{x}}}{x}$ using L'Hopital's rule and series expansion. Do you have other method for solving it?
0
votes
2answers
118 views

Integral of the exponential function

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.
3
votes
3answers
71 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
2
votes
3answers
302 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
2
votes
4answers
49 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
0
votes
1answer
38 views

Formula to link two exponential values together - doesn't quite work

Basically, I've done a script, and I'm stuck on a formula for it. After I run the code on a cube, based on two different inputs (detail level and vertex average iterations), the resulting size will be ...
0
votes
0answers
68 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
2
votes
2answers
70 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
0
votes
4answers
88 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
0
votes
2answers
42 views

help in finding number of solutions of the equation

I wanted to find the number of solutions of the equation: $$3^{(x-1)} + 5^{(x-1)} = 34$$ I can of course find one solution , but how to be sure that there is just one solution.
2
votes
3answers
365 views

Using l'Hôpital rule to find $\lim_{x\to-\infty} xe^x$ [duplicate]

I'm trying to solve this limit: $$\lim_{x\to-\infty} xe^x$$ I'm trying to solve using the l'Hôpital rule. My question is can I use this rule in the last limt below? $\lim_{x\to-\infty} ...
0
votes
1answer
38 views

Solve for coefficients of $y = A(1 - e^{-x/B})$ given two points

I have the equation $y = A(1 - e^{-x/B})$, and two $(x,y)$ pairs. How can I solve for $A$ and $B$? This should be simple, but I've been banging my head against the algebra for a while to no avail. I ...
2
votes
2answers
159 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
1
vote
1answer
115 views

Inverted Circle?

The equation I have is $$\Large x^{\frac23} + y^{\frac23} = 3^{\frac23} $$ I know what the graph looks like, but I don't know how I would find points other than the intercepts mathematically. How ...
1
vote
1answer
30 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
1
vote
1answer
78 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
0
votes
0answers
97 views

Please give me an example of the algorithm where $\Theta$ will be equal to $e^n$

Please give me an example of the algorithm where $\Theta$ or $O$ will be equal exactly to $e^n$ . The algorithm should not be simple counting from 0 till $e^n$ . It should be a clear relation of two ...
0
votes
1answer
23 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
1
vote
2answers
37 views

How Do I set up this problem? continuous compounding

I have no idea how to set up this problem. I am aware of the formula $$A = Pe^{rt}$$ Assume the cost of a gallon of milk is $2.90. With continuous compounding, find the time it would take the cost to ...
1
vote
3answers
62 views

Inverse Laplace Transformation of an exponential function

How one could find the inverse Laplace transformation of $\exp(-(b/(b+s))^k)$? Where both $b$ and $k$ are positive.
1
vote
3answers
72 views

Proving that $ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -…}$

How can one see that for $-1 < u < 1$ we have the following equality $$ 1-u = e^{-u - \,u^2/2 - \,u^3/3 -...} \,\,\,\,?$$ It's probably easy to prove, however I've tried a couple of things so ...
0
votes
1answer
35 views

Normalizing a probability density function

I need to find a normalization term $N(\alpha,\beta)$ for the probability density function: $$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$ In other words, solve the following equation: ...
0
votes
0answers
12 views

Poisson distributed graphs

I am currently reading a paper about poisson distributed graphs and came across the following formula. Apparently the degrees of the graph are distributed binomially through the following ...
0
votes
0answers
55 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
2
votes
2answers
65 views

What do you get when you differentiate a $e^{f(x)}$-like function

I need help with exponential functions. I know that the derivative of $e^x$ is $e^x$, but wolfram alpha shows a different answer to my function below. If you, for example, take the derivative of ...
2
votes
0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
2
votes
0answers
29 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
0
votes
1answer
41 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
1
vote
0answers
82 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...