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

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1answer
73 views

Evaluate the Sum using Exponential Generating Functions

Evaluate the sum $$\sum_{i=0}^{n-1}\binom{i+m-1}{m-1}$$ I know the steps I have to follow but don't know how to apply the process to this example. First I have to find $$A(x)=\sum_{i=0}^\infty ...
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1answer
43 views

Reducing a double summation with infinite limits

I've been solving a Renewal theory problem and I end up with this function $m(t)=e^{-4t}\sum_{k=1}^{\infty}\sum_{i=2k}^{\infty}\frac{(4t)^i}{i!}$. How do I solve or reduce the double summation? Is it ...
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3answers
35 views

Why do I receive the wrong answer when I try to solve this exponential equation?

So I have the equation: $25^{x}=5^{x}+6$ My reasoning is if you make everything to the base 5: $\left( 5^{2}\right) ^{x}=5^{x}+5^{\log _{5}6}$ Given the bases are the same we can do: $2x=x+\log ...
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2answers
212 views

Not commuting exponential matrices

Reading this book I came across the following formula : $$ e^A e^B = e^{A+B}e^{\frac{1}{2}[A,B]} $$ where $A$ and $B$ are two matrices and $[A,B] = AB-BA$. I tried to find a demonstration without ...
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0answers
46 views

Application of exponential distributions

The magnitudes of earthquakes in a region of North America can be modeled by an exponential distribution with mean 2.5 (measured on the Richter scale). If 3 earthquakes occur in a given month, what is ...
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1answer
38 views

What property of summation is used in this simplification?

I am looking at a problem that simplifies $\sum_{n=0}^\infty\frac{(t\mu)^n}{n!}$ to $e^{t\mu}$. I can't seem to recall what property this is. Does anyone recognize this?
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5answers
119 views

Finding the derivative of $2^{x}$ from first terms?

I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) \equiv 2^{x}$, but now I'm ...
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2answers
201 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
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2answers
88 views

Mean and Variance of exponential function

Given this function with j and k as unknown parameters. What is the expression of Variance and Mean of this exponential function? $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 ...
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1answer
67 views

Show that a statistical model belongs to exponential family

I have this statistical model: $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 \pi}}e^{\sqrt{jk}}y^{-\frac{1}{2}} \text{exp}\left( -\frac{1}{2} (j y + \frac{k}{y}) \right) \quad \quad y>0$$ In this case the ...
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2answers
75 views

How to show that exp is a diffeomorphism between symmetric reals and positiv definite matrices?

I am looking for an easy proof of the fact that the exponential function is a diffeomorphism between the finite dimensional vector space of symmetric real nxn-matrices and the open subset of positive ...
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1answer
57 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
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1answer
33 views

Help needed clearing up a textbook explanation of logarithms

A passage in my textbook has me confused, first it states this: $ \log _{a}\left( x\right) .\log _{b}\left( a\right) =\log _{b}\left( a^{\log _{a}\left( x\right) }\right) =\log _{b}\left( x\right) $ ...
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1answer
73 views

Continuity of a map from $\mathbb{C}$ to a Banach algebra

Consider the map from $\mathbb{C}$ to a unital Banach algebra $B$ given by $x \mapsto \exp(xb)$ for $b\in B$. I proved that this map is continuous by using the definition of $\exp(xb)$ as a contour ...
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3answers
77 views

Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$

Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.
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0answers
54 views

Exponential Distribution and Possible Memoryless Property

My attempt : Do you guys think this is right? This might have something do the with the exponential's memoryless property?
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1answer
25 views

Joint distribution proof

I am trying to study for an exam and I am kind of lost on how my professor came to a particular result on his practice exam. Let $W$ be an exponentially distributed random variable with $\lambda = 2$ ...
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3answers
106 views

Conclusion about limit definition of e^a for a sequence of real numbers {a_n} converging to a?

I have seen this fact used in several demonstrations, but have never seen a proof of it. I believe the statement is: If $\{a_n\}$ is a sequence of real numbers such that $a_n \rightarrow a$ finite, ...
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1answer
302 views

Why is the formal solution to a linear differential equation of exponential form?

So $x(t) = e^{ct}$ solves $dx/dt = cx$. This is clear enough from differentiation rules... But I fail to grasp, in some sense which I can't quite put my finger on, why it is so. Why can the solution ...
3
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2answers
77 views

Inverse of the derivative for f(x) = f'(x)

I'm new so forgive my inexperience here. The problem concerns the following: $$ f: \Bbb R \to (0, \infty), f(x) = f'(x) $$ The first part of the problem involves showing f is increasing, this ...
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1answer
41 views

Integral of exponential with second degree exponent

I want to compute the integral $$\int_\mathbb{R}e^{-t\left(y-\dfrac{(at+x)i}{2t}\right)^2}dy$$ I know that $\int_\mathbb{R}e^{-ty^2}dy=\sqrt{\pi/t}$, but here there is an extra imaginary factor. What ...
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2answers
424 views

Finding exact values of $A$ and $B$ from exponential point on graph?

I am faced with a problem that I cannot seem to solve. Here it is: The graph $y = a^x + b$ is shown below, find the EXACT values of $a$ and $b$. The graph is an exponential graph, it has a ...
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8answers
323 views

Given $2^{x}=129$, why is it that I can use the natural logarithm to find $x$?

I've looked at an example in my textbook, it is: $2^{x}=129$ $\ln \left( 2^{x}\right) =\ln \left( 129\right) $ $x\ln \left( 2\right) =\ln \left( 129\right) $ $ x=\dfrac {\ln \left( 129\right) ...
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2answers
49 views

I need some help with the derivative of this function.

Hey guys i was wondering , what is the derivative function of this function. f(x) = $\sqrt{x} - e^{-x}$ Any advise will be greated.
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3answers
71 views

Strange mistakes when calculate limits

I have difficulties with calculating the following limits. W|A gives the correct answers for both of them: $$ \lim_{x \to +\infty} \sqrt{x} \cdot \left(\sqrt{x+\sqrt x} + \sqrt{x - \sqrt x} - 2\sqrt ...
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1answer
31 views

Is the distribution of one exponential will be smaller than a second one Uniform?

I came by an expression which I am not sure I understand. If: $X_1 \sim exp(\lambda)$ $X_2 \sim exp(\lambda)$ Then: $P(X_1<X_2|X_2) \sim Uniform(0,1)$ Where it is not clear to me what ...
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2answers
65 views

Real solutions of the equation $\,z + e^{-z} - x = 0$ for $\,x > 1$ and $\,\Re(z) ≥ 0$

I have the complex equation $z + e^{-z} - x = 0$, where $z$ is complex and $x$ real, and I need to show that it has only one real solution for $x > 1$ and $\Re(z) \ge 0$. I do not see how this is ...
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1answer
59 views

finding an 'e' based limit for this sequence

i need to find the limit for $$\lim_{n \rightarrow \infty} \left(1 + \frac{q}{n}\right)^n $$ where $q \in \mathbb{Q}$ how to i get this sequence to resemble $$\lim_{n \rightarrow ...
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1answer
153 views

Closed form for a fixed point of the exponential function?

Let $$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} $$ denote the exponential function, which is defined on the entire complex plane. There is a fixed point of this function at $w= a+bi$ where $a \approx ...
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3answers
200 views

Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$

Is it trivial to say $$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$ considering the fact that we know $$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$ ...
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0answers
38 views

Why does this method work - exponential functions

We have 2 functions, $f(x) = e^{0.5x-1}$ and $e^{-x+1}$. The line $ y=q$ intersections the functions in P and Q such that $PQ < 3$. Calculate which values for $p$ are allowed. So you get $f(q) = ...
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0answers
113 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
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2answers
106 views

Confusion regarding the Logarithmic function change of base formula

My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated. It starts with: $b^{x ...
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1answer
96 views

How is the gradient of exponential functions with different bases (in the included table) worked out?

I am studying exponential functions at the moment, and this table was presented in my textbook to show that for exponential functions with increasing 'bases' the gradient of the function increases. ...
3
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2answers
125 views

If $x_n = (\prod_{k=0}^n \binom{n}{k})^\frac{2}{n(n+1)}$ then $\lim_{n \to \infty} x_n = e$

I try to prove the following: $$x_n = \left(\prod_{k=0}^n \binom{n}{k}\right)^\frac{2}{n(n+1)}$$ $$\lim_{n \to \infty} x_n = e$$ I want to use double sided theorem, so I've proven that $$x_n \ge ...
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1answer
114 views

Natural growth and decay rate of a bacteria culture

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $420$. (a) Find an expression for the number of bacteria ...
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2answers
1k views

What is the difference between logarithmic decay vs exponential decay?

I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.
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1answer
66 views

How to find a rule from a table

X | Y 1 | 18 2 | 24 3 | 42 4 | 96 How does one find a rule from a table like this? The only way I am able to find rules to ones like this is by using guess and check but I know there must be a ...
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1answer
91 views

Why does $e$ seem to be an intuitive number? [closed]

I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would ...
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4answers
167 views

$\lim _{ n\rightarrow \infty }{ { \left( -1+\frac { 1 }{ n } \right) }^{ n } } =? $

We know that $$\lim _{ n\rightarrow \infty }{ { \left( 1-\frac { 1 }{ n } \right) }^{ n } } =\frac { 1 }{ e } .$$ However the result of $$\lim _{ n\rightarrow \infty }{ { \left( -1+\frac { 1 }{ n ...
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1answer
79 views

Practical significance of $e$ [duplicate]

We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't ...
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1answer
148 views

Proofs that there is no $f(z)$ such that $\exp f(z) = z$ for all $z \in \Bbb{C}\setminus\{0\}$

When I first learned about this result I was completely stunned that there is no holomorphic function $f(z)$ on $\Bbb{C}\setminus\{0\}$ such that $\exp f(z) = z$. What are some interesting proofs of ...
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1answer
53 views

Calculating cumulative Markov Chain outcomes

I have a Markov process, with 2 possible states (1 or 0) and a transition matrix P. State at time t=n is determined by x0*Pn. As n goes to infinity, xn goes to the steady state vector, q = [q1 q2]. ...
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3answers
54 views

solve the exponential function

solve $(\sqrt6-\sqrt5)^x-\sqrt6-\sqrt5=0$ I know the result is $-1$ but I don't know how to prove it. I have tried to replace $\sqrt6-\sqrt5=t$ but then I have $t^x-t+2\sqrt5=0$ and I think it is ...
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2answers
134 views

Why is it important to define that a logarithm and exponential function is one-to-one?

I'm currently studying the properties of logarithm in an open source pre-calculus textbook that can be found here (Page 438). Before the text goes on to the Algebraic properties of exponential and ...
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2answers
41 views

Exponent multiplication error

$\left( x^{2/3}\right) ^{3/2}=x^{\dfrac {2}{3}\times \dfrac {3}{2}}=x^{1}$ Given this why is it that if I substitute $x=-1$ I get 1?: $\left( \left( \sqrt [3] {-1}\right) ^{2}\right) ...
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1answer
77 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
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2answers
57 views

Changing an exponential function to logarithmic

I have a question stating that $P=75e^{-0.005t}$ and they want to get t by itself. I used the example $y=2^x = x=log_2(y)$ To find that $-0.005t = 75ln(P)$ So $t=\frac{75ln(P)}{-0.005}$ However ...
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1answer
48 views

Help with solving $\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$

Given $$\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$$ Can you please show the step(s) involved to reach this next line in the textbook: $$\left[-\frac 1s\exp\left(\frac sr e^{-rt} \right) ...
1
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1answer
368 views

Find the Maximum and Minimum values of $e^z$ when $z\le 1$.

I need help finding the maximum and minimum values of $|e^z|$ on $|z|\le1$. I know we use the maximum modulus theorom but i cant seem to get an answer.