Questions about exponentiation

learn more… | top users | synonyms (1)

1
vote
0answers
102 views

Collaborative modular exponentiation

EDIT: Rephrased. I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$. I also know $b$, $R$ and $N_2$ (but not its factors). I want to know whether there is ...
1
vote
1answer
101 views

How to formulate exponential growth?

Here's my question: A rumour spreads exponentially through a college. 100 people have heard it by noon, 200 by 1pm. How many people have heard it a) by 3pm b) 12.30pm c)1.45pm ...
0
votes
1answer
185 views

Solve equation with variable in exponent as well as base

I'm having a bad brain blockade right now... I'm trying to solve the following equation for $n$: $$PV=\frac{FV}{\left(1+\frac{r}{n}\right)^{nt}}$$
2
votes
1answer
793 views

Exponentially Distributed Random Variable - P.D.F

Here's an interesting question: The area of a circle is exponentially distributed with parameter $\lambda$. Find the distribution of PDF of the radius of the circle. Then assume that the ...
11
votes
1answer
446 views

Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function? ...
0
votes
1answer
220 views

Approximate solution for an exponential equation

Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P: $$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$ Where M, N are parameters. The ...
2
votes
2answers
71 views

What is the math behind this transformation on exponents that are logarithms?

I understand that $$a^{\log_b(n)} = n^{\log_b(a)}.$$ What is the math behind this transformation that allows you to swap the $a$ and $n$?
0
votes
3answers
233 views

Get the number, knowing the power and result of raising.

If I have a number and a power of this number. How to get the number that raised to the known power will give the known number? $$x^3 = 90$$ How to get $x$?
2
votes
4answers
266 views

What is the meaning of $\exp(\,\cdot\,)$?

What is the meaning of the notation $\exp(\text{expression})$ ? I think that it's something of the form $a^\text{expression}$ but does it mean that the base $a=e$ or can it be any base?
1
vote
1answer
40 views

Formula to scale a series that is being bent with a root / power.

I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
3
votes
0answers
81 views

Numerically estimate $a^b$ [duplicate]

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like ...
4
votes
1answer
359 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
5
votes
2answers
259 views

How can I solve $x^x = 5$ for $x$? [duplicate]

Possible Duplicate: Is $x^x=y$ solvable for $x$? I've been playing with this equation for a while now and can't figure out how to isolate $x$. I've gotten to $x \ln x = \ln 5$, which seems ...
107
votes
9answers
4k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
0
votes
2answers
698 views

Graphing Fractional Exponents

$f(x)=x^\frac{5}{3}-5x^\frac{2}{3}$ is the same as : $f(x)=(\sqrt[3]x)^5-(\sqrt[3]{5x})^2$ Except, with the first equation, my calculator returns an error for negative values of $x$ (We are ...
6
votes
1answer
409 views

Is it known or new? [duplicate]

Possible Duplicate: Starting digits of 2^n While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact. My ...
1
vote
1answer
98 views

I noticed a pattern, does this have a name?

First of all I am a programmer, not a mathematician, so I may articulate what I am trying to say very poorly. I was working with powers of $2$ when I noticed a relationship I had never noticed before. ...
6
votes
4answers
5k views

Solving a sum of exponentials

If I have an equation of the form $$e^{ax} + e^{bx} = c,$$ where $a$, $b$, and $c$ are constants, how can I simplify the equation to solve for $x$? Taking the logarithm of both sides is tricky, since ...
4
votes
4answers
278 views

Find $n$ where $n^n$ has $n$ digits

Find n $\in \mathbb{N}$ if $n^n$ has $n$ digits. A problem I ran into today and it seemed interesting. I know $1$, $8$ and $9$ are (the obvious) solutions, but are these the only ones? If they are, ...
1
vote
5answers
171 views

How can I find $c=d^x$ using only basic computer functions?

Since this seems more like a math question than programming, I'm placing it here instead of on SO I'm trying to find $x$ when $c$ and $d$ are known constants (user input and iteration counter, ...
4
votes
2answers
197 views

powers of 2 in base 3

What is the asymptotic limit of the ratio of $1s$ to $2s$ in the first digits of $2^n$ in base $3$? If the $2^{nd},3^{rd}$ digits etc. were random but equally likely to be $0,1$ or $2$ then the ...
7
votes
2answers
455 views

“Wild” exponents of $e$

One of the things I'm curious about is why do some functions describe something like this: $$f(x,y) = e^{-\frac{x^2+y^2}{2\sigma^2}}$$ And people mostly take it for granted, throw it around for ...
68
votes
5answers
2k views

Root Calculation by Hand

Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please? Well, when I use a Casio scientific ...
3
votes
3answers
231 views

What is the fractional part function of $e^x$?

Given a real positive number $x\in\mathbb{R^+}$. What is the function of the fractional part of $e^x$?
1
vote
2answers
102 views

How to simplify the product of two $\exp$ functions

It's been a while since I did any of this. I have the following product: $\exp(-j2 \pi u|k|x) \cdot \exp(-j2 \pi v |k|x)$. This seems like it is something that can be simplified, but how? Note, ...
1
vote
3answers
4k views

Multiplying exponents with variables inside

Why is $$(-1)^n(2^{n+2}) = (-2)^{n+2} ?$$ My thinking is that $-1^n \times 2^{n+2}$ should be $-2^{2n+2}$ but clearly this is not the case. Why is the variable essentially ignored, is there a ...
4
votes
3answers
358 views

How do I find if $\frac{e^x}{x^3} = 2x + 1$ has an algebraic solution?

Is there some way of solving $$\frac{e^x}{x^3} = 2x + 1 $$ non-numerically? How would I go about proving if there exists a closed form solution? Similarly how would I go about proving if there exists ...
1
vote
1answer
126 views

Convert natural exponent, $e^{c\cdot x}$, into the form $a^{x}$

How does one convert a natural exponent written as $e^{c\cdot x}$ into the form $a^{x}$ ?
5
votes
1answer
212 views

Is there a definite integral that yields $e^\pi$ or $e^{-\pi}$ in a non trivial way?

The title says it all. No trivial answers like $\int_0^\pi e^tdt$ please. The idea is rather, if there are integrals like $$\int\limits_0^\infty \frac{t^{2n}}{\cosh ...
3
votes
2answers
66 views

How to calculate $(k^x)^{-1} \pmod m$

I'm trying to follow this example of ElGamal's encryption scheme (page 2, slides 3 & 4), but don't understand this step (decryption, step 2): $$(k^x)^{-1} \pmod m$$ Where $$k = 10,\ x = 9,\ m = ...
1
vote
3answers
191 views

Exponential norm?

Can a norm "grow exponentially"? Let $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0} $ be a norm such that: $$ \lim_{|x| \rightarrow \infty } \frac{ ||x||_* }{ e^{|x|} } > 0 $$ where ...
-4
votes
4answers
308 views

How to derive a formula for the first n powers of an integer?

How to derive a formula for the first n powers of an integer? In particular, sum of 2^n? I'm looking for a proof that not only utilizes algebraic manipulation but is also easily seen visually.
1
vote
2answers
907 views

Find all solutions to the equation $e^z = i$

I know that there is an equation for finding the nth roots of a complex number, which easily done once you have the modulus and argument of the complex number in question. There would be n roots. But ...
1
vote
3answers
937 views

Modular exponentiation?

I came upon an interesting way to relatively quickly compute modular exponentiation with large numbers. However, I do not fully understand it and was hoping for a better explanation. The method ...
4
votes
2answers
146 views

Is the multiplicative complex plane a Lie group?

I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication? This would give us a nice geometrical interpretation of the famous Euler ...
4
votes
1answer
891 views

Comparing Powers with Different Bases Using Logarithms?

I looked all over to see if a question like this had already been answered, but I couldn't find it. So here goes: I need a general formula for comparing two (insanely huge) powers. I'm pretty sure ...
1
vote
1answer
136 views
0
votes
1answer
101 views

Solution to $x^{\frac{1}{x}} = 0$ using Lambert's W?

In a previous question, I learned that the equation $$x^{\frac{1}{x}} = c$$ have no solutions when $c = 0$. Below, I tried using Lambert's W function, and I found a solution at $x = 0$. Did I ...
0
votes
2answers
214 views

Solve $x^{\frac{1}{x}}=0$

When answering this question, I contemplated across the following problem: Find $x$ such that $x^{\frac{1}{x}}=0.$ I thought RHS $ = 0$ is a special case, so I attempted solving $$\frac{1}{x} ...
105
votes
3answers
8k views

Is 2048 the highest power of 2 with all even digits (base ten)?

I have a friend who turned 32 recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being 32 was pretty good since not only is it even, it also has no odd factors. ...
0
votes
2answers
56 views

Choices for Integrationg by Parts

I am working on an integration by parts problem, and, it looks like I got something incorrect somewhere. I have been told to follow LIATE (L ogarithmic, I nverse trigonometric, A lgebraic, T ...
0
votes
2answers
205 views

Simplifying exponents, multiplication, and addition

How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
0
votes
1answer
92 views

Negative even exponents of negative numbers

How can I calculate and prove this equation with mathematical terms: pow((-2), -2)=? I know that pow(1, -1) is equal to 1/1 by the way. Any idea, please?
3
votes
2answers
2k views

Growth of exponential functions vs. Polynomial

Will $2^x$ take over $x^{1000}$ ? I thought that exponential functions had the fastest growth rate, however, graphing it on wolfram alpha made it seem as if the initial behaviors of the two functions ...
2
votes
2answers
328 views

Is there a reasonable generalization of the falling factorial for real exponents?

The falling factorial is defined as: $$x^\underline n = \prod_{k=0}^{n-1}(x-k),\quad n\in\mathbb N$$ It can be used to define the binomial: $$\binom n k = \frac{n^\underline k}{k!}$$ And it ...
2
votes
3answers
177 views

Mathematical function for the powers

I have this formula $$\underbrace{2^{2^{2^{.^{.^{.^{2^2}}}}}}}_n$$i.e. where the total number of 2's is $n$. Is there any way to write it as a single mathematical function?
2
votes
1answer
148 views

Has a matrix block diagonal structure if and only if its exponential has it as well?

Obviously if $\mathbf{A}=\begin{bmatrix}\mathbf{C} & \mathbf{0} \\ \mathbf{0} & \mathbf{D}\end{bmatrix}$ then $e^{A}=\begin{bmatrix}\mathbf{e^C} & \mathbf{0} \\ \mathbf{0} & ...
1
vote
1answer
109 views

The power of a power

My teacher gets the following: $(x^{2})^{12-k} = x^{2k-24}$ Where I get the following: $(x^{2})^{12-k} = x^{24-2k}$ I'd like to think of $2(12-k)$ as $2*12 - 2*k$ or $-2k + 24$. Why/how am I wrong? ...
0
votes
2answers
706 views

Notation of inverse trigonometric functions and exponentiation [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ I have worked a bit on trigonometry today, and something strikes me as inconsistent. In the book, the notation for the inverse sine ...
21
votes
1answer
2k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...