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

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1
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1answer
45 views

Find real-valued sequences $x(n)$ for which $c^{x(n)} = o(1/n )$

For which $x=x(n)$ does it hold that $$c^x = o\left(\frac{1}{n}\right)$$ where $c\in(0,1)$ is a constant. So clearly, for $x=n$, this is true. But for which $x =o(n)$ does this hold? I thought ...
6
votes
1answer
370 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
0
votes
1answer
29 views

How to scale a equation e.g. by log

I'm currently trying to scale an equation since the numbers I have to calculate with are pretty large and Matlab outputs Infinity (Inf). However, the question here is more about the mathematics behind ...
0
votes
2answers
54 views

Construction of complex numbers and exponent rules for them

I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers. $1$. Is it enough to define it as $a+bi$, where a,b are ...
-1
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4answers
47 views

Exponential function (t)

I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.
0
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1answer
55 views

How can we know that x^x is an exponential function or not without drawing the graphic?

In general, exponential function is defined as a.b^x, where a=coefficient, and b= base. I only knew that the function is exponential function or not, just by drawing the graphic. But, how can we ...
4
votes
1answer
57 views

Where do I make mistake on this derivative containing e^x^2

My brother is preparing for the university and asked me the following multiple choice question. $$\frac{d}{dx}(x^3 * e^{x^2})$$ a) $e^{x^2}*x^2*(1+2x)$ b) $e^{x^2}*x^2*(3+2x)$ c) ...
0
votes
1answer
20 views

Explicit and Recursive Exponential Growth

The population of a certain organism triples every hour. Write a function that models this growth. By what factor does the population grow in one-half hour? I'm unsure of how I should approach the ...
2
votes
4answers
67 views

How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r ...
1
vote
1answer
48 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
0
votes
0answers
30 views

Log(x) or Exp(x) with limits 0,0 and 1,1

trying to find an equation or function that displays an accelerated progression. It may be an Exp or Exp-like function. Input is X, ranging from 0 to 1. Output is Y ranging from 0 to 1. Note Y must ...
2
votes
4answers
80 views

Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
1
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2answers
37 views

What kind of mathematical operation is used to repeatedly increase a number by a certain percentage?

I am sure that this is an easy question to answer for most of you. I need to take a number, let's say $10$, and then increase it by a percentage, let's do $25\%$. $10 \times 1.25 = 12.5$ Easy ...
2
votes
1answer
81 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
2
votes
2answers
62 views

Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not ...
0
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0answers
14 views

Explicitly relating two functions containing exponential terms

I have two functions related to the distribution of administered drugs in the body: $$\begin{align}c_1(t) &= a_1\exp(-k_{11}t) - b_1\exp(-k_{21}t)\\ c_2(t) &= a_2\exp(-k_{12}t) - ...
1
vote
5answers
85 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
53 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
100 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
38 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
51 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
95 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
194 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
140 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
328 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
68 views

Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
1
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2answers
61 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
68 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
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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 ...
0
votes
2answers
59 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
92 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
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0answers
40 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
92 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
72 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
101 views

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

Define the function $g_n(z)=(1+\frac{z}{n})^n$ for $\:n\in \mathbb{R^+}$. Then $\frac{d}{dz}g_n(z)=n(1+\frac{z}{n})^{n-1}\cdot\frac{1}{n}=(1+\frac{z}{n})^{n-1}$ Define $g_{\infty}(z)=\lim ...
2
votes
1answer
22 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
39 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
255 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
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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
295 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
46 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
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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
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0answers
61 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
68 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
87 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
36 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 ...