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

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2answers
41 views

exponential regression for bacteria growth

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this: $$(0,32), (2,65),(...
0
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2answers
65 views

Can you easily simplify these terms?

I need to simplify these terms step by step to prove they are equivalent $$(100^{(2n+1)}-1+99×10^{(4n+2)}-99×10^{(2(n+1)-1)})/(11×10^{(2n+1)}-11) $$ and $$(100^{(2(n+1)+1)}-1)/(10^{(2(n+1)+1)}-1)×1/...
1
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1answer
100 views

prove that the following function is decreasing?

I am trying to prove that the following function is decreasing. \begin{align}&f(t)=\frac{1-g(t)}{\sqrt{1+e^t}}\cdot\exp\left(-\frac{te^t}{2(1-e^t)}\right)&t<0\end{align}where $ g(t)=\dfrac{(...
1
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1answer
84 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so $$V=A\cos(x)+B\sin(...
-1
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2answers
67 views

Integral of $\int\frac{1}{1+2e^x}dx$

It seems there are two ways to find the integral of this function $f(x) = \frac{1}{1+2e^x}$. In both paths I only do operations that I know are true, but for some reason one of them gives me the right ...
0
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1answer
68 views

solving for a variable that exist inside as well as outside of natural log or exponent

can the following equation be solved for K analytically? If not, then what other approaches I could try out? K*ln[(C2-K)/(C1-K)] = -(F/V)*t The original equation ...
0
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1answer
43 views

Complex numbers converge if their absolute values and arguments converge

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$. My proof: $z_n= |z_n|e^{i \arg(z_n)} \...
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2answers
58 views

Closed form of $\int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \mathcal{I} = \int_{\delta_1}^{\delta_2}(1+Ax)^{-L}x^{L}\exp\left(-Bx\right)dx, \end{equation} where $A$, $B$, $\...
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2answers
58 views

$e^{\mathrm{Re}\,z}$ not analytic in complex plane

In my textbook I found a text where it says that $e^z$ (z is a complex number) is analytic everywhere. But $e^x=e^{\mathrm{Re}\,z}$ is not. How can I prove that about $e^x$ and what is the ...
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9answers
347 views

Why $e^x$ is always greater than $x^e$?

I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the ...
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0answers
28 views

Evaluate the area of the intersection of two amoebas

Here is defined what is a amoeba and what is a tentacle, see page 3. I believe that this could be a nice exercise in multivariable calculus. Question. Can you give an example to show how compute ...
-1
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1answer
38 views

Solving an integral with exponential function

I try to solve the following integral $$\int_a^b \exp\left\{-\lambda \left(\frac{y}{2x^2}-\frac{1}{x}\right)\right\} dx$$ for $\lambda>0$ and $y \in \mathbb{R}$. Do you see any relation to any ...
1
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3answers
66 views

Exponential of a square matrix [closed]

I need to find the matrix exponential $\exp(At)$ where $$A= \begin{pmatrix} -a & 0 \\ 1 & 0 \\ \end{pmatrix}. $$
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2answers
141 views

Simplify $(2^{2015})(5^{2019})$

Question : $(2^{2015})$$(5^{2019})$ How do I simplify/solve that without a calculator? I have no idea how to continue, I know it's important to get the Base number the same so I can add the ...
1
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0answers
57 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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1answer
22 views

Evaluating an integral that contains a positive definite matrix

Let $A \in \mathbb{R}^{d \times d}$ be a positive definite matrix (meaning that $z^t A z > 0 \space \forall z \in \mathbb{R}^d, z ≠ 0$, although I'm also allowed to use any other of the "usual" ...
2
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1answer
36 views

Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...
0
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0answers
15 views

Which one is the right answer if this was on the SAT Math IIC?

If $300 is invested at 3%, compounded continuously, how long (to the nearest year) will it take for the money to double? (If P is the amount invested, the formula for the amount, A, that is available ...
2
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0answers
34 views

Does the kind of singularity of a meromorphic function $f$ determines the kind of singularity of $e^f$? [duplicate]

In this question, I recently asked on if the isolated singularities of a meromorphic function $f$ are the same ones as the of the function $e^f$. I quickly realized that this isn't the case, so I ...
0
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2answers
111 views

Can we interchange the limit and integration for $\int f_n(x)\,\mathrm{d}x$? [duplicate]

In connection with this question about computing integrals of the form $$\mathcal{I}_n=\int_0^\infty f_n(x)\,\mathrm{d}x=\int_0^\infty \frac{\mathrm{d}x}{e^x+x^n}$$ I noticed an interesting trend ...
0
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1answer
22 views

Are the isolated singularities of a holomorphic function $f$ from the same kind as the ones from $e^f$?

Given any meromorphic function $f: U \to \mathbb{C}, U \subseteq \mathbb{C}$ open, I want to prove or disprove the statement: for any isolated singularity $a \in U$, the function $g(z) = e^{f(z)}$ has ...
2
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2answers
42 views

Is it more accurate to use the term Geometric Growth or Exponential Growth?

On Wikipedia, the terms Exponential Growth and Geometric Growth are listed as synonymous, and defined as ...
1
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7answers
58 views

Solving exponential system

How can I solve the following system? \begin{cases} x^y = 16 \\ \frac{x}{y}=2\\ \end{cases} I tried everything, nothing works.
2
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1answer
51 views

Solving this exponential

I'm trying to solve $$ e^\lambda (1-\lambda^2) = 1$$ I know it has a solution at $\lambda = 0$. How do I get the second solution? Wolfram Alpha gives something around 0.71, but doesn't show an ...
0
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1answer
38 views

Money problem…probability spend in particular time

A child puts money in piggy bank every day , in particular 10 , 20 , 30 , 40 , 50 , or 60 cents with the same probability . Find the probability of spending at least 80 days before having collected 30 ...
0
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1answer
23 views

finding the probability of not being answered more than two calls!

In a call center , the time between successive calls is exponential random variable with $\mathbf{E}X$= 2 minutes . If the operator is removed for 5 min , what is the probability of not being answered ...
1
vote
4answers
80 views

How do I show that $-\frac{1}{e^x + 1} + 1 = \frac{e^x}{e^x + 1}$?

The expression is $$-\frac{1}{e^x + 1} + 1 = \frac{e^x}{e^x + 1}$$ I would like help to get from the left side to the right side.
1
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2answers
68 views

How much of radioactive material substance will remain after $5$ days?

Initially there are $8$ grams of a radioactive material in a container.The half-life of the material is $2$ days. How much of the radioactive substance will remain after $5$ days ? By exponential ...
1
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2answers
56 views

Tangent line Exercise. $f (x) = 5e^{−(x−2)^2}$ . Find the coordinates of points where the hill is the most steep.

I have this exercise, look easy but I don't know where to start, I think that I need a extra function or value to continue with the calculus. Imagine that you are riding over a hill having its ...
1
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0answers
55 views

Proof of the inequality $e^x \le x+ e^{x^2}$ [duplicate]

$e^x \le x+ e^{x^2}$ for $x\in \mathbb{R}$. I'm struggling to show this inequality. I've tried differentiation but to no avail. How can I show this? I would greatly appreciate some help.
1
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1answer
26 views

Derivative of the inverse of exponential function a^x, with a>0 and a≠1

While studying exponential functions, I understood that $$\frac{d}{dx}a^x=(\ln a)a^x.$$ I also learned previously that if $g(x)$ is the inverse of $f(x)$, then the derivative of $g(x)$ and the ...
0
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3answers
61 views

Solving an exponential equation with absolute value

I am to solve this equation: $|\frac{-2^x}{1-2^x}| < 1.$ And so I got rid of the modus sign: $\frac{-2^x}{1-2^x} < 1 $ or $\frac{-2^x}{1-2^x} > -1$ But I am stuck now. How should I ...
0
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5answers
60 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = \frac{...
1
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2answers
77 views

A circle centered at the origin is tangent to $y=2^x$. What is the radius of the circle?

I feel as though I am doing the analytical part correctly, however, where I am facing the roadblocks in this problem is in the actual algebra itself. Perhaps I am not doing something right in my ...
1
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5answers
84 views

how do you solve exponential equations with added bases

im confused on how you solve a question like this: $$ 3^{x+2} + 3^{x-1} = 27 $$ would you do: $$ 2(3)^{2x}-1 = 3^3 $$ but when I try this way its wrong, please help me thanks.
0
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1answer
114 views

If $f(z)$ is meromorphic but not entire, is $\exp(f(z))$ meromorphic? Could it even be entire?

First, I can show that $f$ meromorphic is a rational function. Now, I want to consider $g=e^{f(z)}$. I have heard that there is something interesting that goes on with $g$, that there is some room ...
0
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1answer
57 views

Limit of the Exponential Integral

I want to show the following (I am only really interested in real variables/parameters, so $x,b,c\in\mathbb{R}$): $$\lim_{x\rightarrow\infty}E_1\left[b\left(x+c\right)\right]=0,\quad\text{for }b>0$...
6
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0answers
255 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
0
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0answers
44 views

Exponentiated Operators? ($e^{\hat{A}+\hat{B}} \ne e^{\hat{B}+\hat{A}}$)

Given, $$ e^{\hat{A}+\hat{B}} = e^{\hat{B}}e^{\hat{A}} $$ I then consider the series expansion of both exponentials. This then leads to a particular order of operation derived from the order of ...
6
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5answers
667 views

Solving exponential equation with unknown on both sides

I am having trouble solving the equation $$3e^{−x+2} = 5e^{x-1}$$ Any help would be appreciated. Thanks.
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2answers
88 views

is there any integral from zero to infinity that sums up to e? [closed]

This may be very basic question, I just don't know. I just want to look and study that function. Thanks in advance Edit: Sorry it may not meet requirements of a good question. I probably don't know ...
0
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1answer
32 views

Combinatorics problem using generating exp functions 2

Calculate the number of sequences of length n that are made of $1, 2, 3, 4$ so that the digits $1,2$ shows an even number of times, And the digit $3$ shows at least 1 time. I've been given a clue to ...
1
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1answer
53 views

Combinatorics problem using generating functions 1

In how many ways can you divide $n$ different balls into $5$ different boxes so that the two last boxes has an even number of balls. I've been given a clue to show that: $\sum_{n=0}^\infty {x^{2n} \...
2
votes
3answers
118 views

Why does the minimum value of $x^x$ equal $1/e$?

The graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for ...
1
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2answers
15 views

describing a decay process with exponentials and differential equations

I have a process of degradation of some material that proceeds like this across time $t$: $C_t = C_{t-1} + RC_{t-1}$ where $C_t$ is the amount of material at time $t$ and $R$ is a (negative) rate of ...
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2answers
36 views

Algebra of exponential

Solve for $x$ in exact value: $\\3^{2x}-3^{x+2}+8=0$ I have tried substituting $3^x$ $=a$ but I didn't get anywhere. $\\a^2-a^{1+\frac{2}{x}}+8=0$
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2answers
38 views

Solving exponent on both side of equation

I'm new here on Mathematics and have only basic algebraic knowledge. I have a problem in how to solve the following equation: $$ P^x = R_0^x + R_1^x + ... +R_n^x $$ I know the value of P and the ...
0
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1answer
85 views

exponential martingale inequality

I stumbled across a claim I couldn't verify. Let $M_t$ be a continuous local martingale, $M_0=0$ a.s. and $\lambda>0$. Then $$ \mathbb{E}\left( \exp \left( \lambda M_t \right) \right) \leq \sqrt{\...
1
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3answers
35 views

Transformation of exponentials

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried: Let $y=3^x$ to $y=e^{x'}$ $$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$ $$x\...
1
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0answers
62 views

How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...