# Tagged Questions

1answer
67 views

### How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x …$ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ...$ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
1answer
35 views

### Big-Oh of exponent of exponent

How does one whether an exponent of an exponent is the big-Oh of the other? For example, if I have $a^{b^n}$ and $b^{a^n}$, how would i determine and prove which is a big oh of another? I'm thinking ...
2answers
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### Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
3answers
68 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 ...
1answer
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### 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 ...
1answer
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### $\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
1answer
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### Something that grows faster than NP class of problem does

I have a theoretical question. F.ex. we have a NP-class of problem, i.e. which do need exponential time on deterministic Turing machine. Is there anything that is growing faster than exponent does. ...
1answer
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### Question about calculating exponent of polynomial

$V=R_{3}[X]$ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
1answer
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### Why is $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$?

I came across this statement, but can't see why it holds: $\left(1-\frac{1}{k}\right)^t < e^{-t/k}$ I'm sure it's something simple, but I don't have a great deal of mathematical experience. I ...
1answer
30 views

### Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
4answers
84 views

### What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
1answer
48 views

### Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
2answers
33 views

### Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
1answer
67 views

### how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
1answer
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### On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0answers
21 views

### Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
2answers
32 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 ...
0answers
15 views

### How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
1answer
55 views

### $|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
7answers
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2answers
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### Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
4answers
146 views

### Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
1answer
60 views

### A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
1answer
187 views

### Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
0answers
49 views

### Are the solutions to this integral known?

Mathematica knows that the real part of $y$ in this integral: $$\int_0^{\infty } \frac{1}{x^{1/y}+2} \, dx$$ is: $$0<\Re(y)<1$$ Therefore I am wondering if the solutions to this integral known? ...
0answers
44 views

### Exponential equations solving methods?

Do you have an idea or general method to solve the following equation?: $$a^{\alpha x}+b^{\beta x} = c^{\gamma x}+ d^{\delta x}$$ when $a,b,c,d$ aren't zero, and $\alpha, \beta, \gamma, \delta$ are ...
1answer
90 views

### Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
3answers
80 views

### Complex exponential

I know that the equation $e^{z}=-1$ has no solution had if been $z$ is a real number. So does the equation also has no solution when $z$ is complex?
2answers
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1answer
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### Why do we use the form $f(t)=ae^{kt}$ for exponential growth and decay?

Why do we include the $e^{k}$? Wouldn't it be easier to simply use $f(t)=ap^{t}$ where $p$ is the percentage increase per time. Is there a reason why the convention is to use $f(t)=ae^{kt}$?