Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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21
votes
5answers
3k views

Fun logarithm question

How would you go about solving $x^{2^{x}} = 2^{x}$? There should be a solution $1<x<2$, but I haven't found a way to derive the answer using the usual log laws, maybe there is an elegant way ...
1
vote
1answer
136 views

If $b > 1$ and $B(r)$ is the set of all numbers $b^t$, where $t$ is rational and $t \leq r$, prove that $b^r = \sup B(r)$ where $r$ is rational.

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. As a follow-up to this question, Rudin asks ...
2
votes
0answers
55 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 ...
1
vote
1answer
120 views

Matrix to Matrix Power [duplicate]

Considering $A^b$, we have a nice definition and properties when $A$ is a matrix and $b$ is a real (or field) value. We also have a fairly useful definition for $e^B$, where $B$ is a matrix and $e$ ...
0
votes
2answers
121 views

Prove $(a^b)^c = a^{bc}$

I need to prove the exponent identity $(a^b)^c = a^{bc}$, where $a,b,c \in \mathbb{Z}$. Apparently this proof is elementary/trivial, but I can't think of how to prove it. I need it as a lemma for ...
0
votes
1answer
54 views

Determine Exponential Equality Without Calculating Values

I want to determine the set of equivalent values in exponential form based on a series of bases and powers. For example $2^k$, such that $2 \leq k \leq 100$ (ie, $2^{2-100}$) compared to $4^k$, ...
1
vote
3answers
76 views

Representing Complex Exponentials with Real and Imaginary Parts

My confusion lies with this : http://www.wolframalpha.com/input/?i=modulus+%28cos%282+pi+r_1%29%2Bcos%282+pi+r_2%29%2Bi+%28sin%282+pi+r_1%29%2Bsin%282+pi+r_2%29%29%29+squared I was looking at ...
0
votes
3answers
259 views

Prove that $(b^m)^{1/n} = (b^p)^{1/q}$ if $r = m/n = p/q$

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. The problem states that $b > 1$ and $m, ...
4
votes
2answers
118 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
6
votes
3answers
238 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
5
votes
1answer
268 views

Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

$SO(3)$ denotes 3x3 rotation matrices. This is Lie group, with corresponding Lie algebra being $\mathrm{Skew}_3$, the space of 3x3 skew-symmetric matrices. The link between them is the matrix ...
1
vote
0answers
43 views

Rules for $(x^a)^b$ - complex number. [duplicate]

How potentiate $(x^a)^b$ for complex numbers?
6
votes
6answers
324 views

Why does $a^{b}·a^{c}=a^{b+c}$?

Why does $a^{b}·a^{c}=a^{b+c}$ ? I want proof for it, I asked professor and he replied: "I don't know, it's a property anyway that is true and that's all what you need to know."
6
votes
2answers
176 views

Riemannian geometry: …Why is it called 'Exponential' map?

The exponential map $exp_{p}:T_{p}M \to M$ given a suitable $v \in T_{p}M$, returns $p$, displaced along the geodesic uniquely determined by $(p,v) \in TM$ for unit "time". So, what does the above ...
2
votes
3answers
148 views

Modular Arithmetic with Powers and Large Numbers

I have a question similar to the following: Evaluate $8767^{2123} \mod 15$. So I got $(8767^{11})^{193}$ $(8767^1*8767^{10})^{193}$ $(8767^1*(8767^3 *8767^4))^{193} \mod 15$ Now I haven't ...
0
votes
1answer
97 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 ...
4
votes
2answers
288 views

Calculating $a^n\pmod m$ in the general case

It is well known, that $$a^{\phi(m)}\equiv1\pmod m ,$$ if $\gcd(a,m)=1.$ So, $a^n\pmod m$ can be calculated by reducing n modulo $\phi(m)$. But, for the tetration modulo $m$ $$a \uparrow ...
11
votes
0answers
135 views

Biggest powers NOT containing all digits.

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$ Consider the powers $m^n$ , for which there is at least one digit not occurring in the decimal representation. Is there a largest $n$ ...
1
vote
1answer
197 views

Showing (1 - polynomial fraction) raised to a polynomial power is a negligible function

Let $P(k)$ and $Q(k)$ be two polynomials ($k>0$). Let $\mathrm{neg}(k)$ be a negligible function for sufficiently large $k$ (see Appendix on question for definition). Does someone know how to show ...
0
votes
3answers
91 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?
1
vote
1answer
107 views

Find $x$ in $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot …] x)^{ (((1\cdot x+1)x +1)x +1)x+1…} $

If $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot ...] x)^{ (((1\cdot x+1)x +1)x +1)x+1...} $, and $\large \,\,A = (\sqrt[3]{3\sqrt 3 })^{\frac{\sqrt 3}{3}} $, find $x$ I have ...
0
votes
4answers
87 views

Calculate approximately the expression $A = 5^{1/2} . 5^{1/4} . 5^{1/8}…$

Calculate approximately the expression $A = 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8}\cdot\ldots$ My books says to says to do this: $ 5^{1/4} \cdot 5^{1/8} \cdot 5^{1/16}\cdot\ldots = A $ Then $ A = ...
2
votes
2answers
111 views

Definitive answer to existence of real exponents

Well, I've been searching through this fórum and I know this question has been answered many times. But the answers I see, are kinda circular (I think). Let's start by the natural case. Natural case ...
2
votes
0answers
83 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
0
votes
1answer
43 views

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; \,\, a^{1/a} > b^{1/b}$

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; $ $$\,\, a^{1/a} > b^{1/b}$$ I need some tip to start it. Thank you.
0
votes
2answers
1k views

Evaluate a matrix with a negative power

I am having problem with how to calculate a matrices that are raised to negative powers. I can manage the adding, multiplication etc, but I am stuck here. The matrix in question is ...
5
votes
2answers
139 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
2
votes
2answers
95 views

Differentiation of exponential function? [closed]

How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $$ {}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ...
0
votes
3answers
134 views

Find $x ^{ 2013} + 2013x ^{ 2010}$

Q. If $\large {\space x^2 + x + 1 = 0\space } $, Find $ x ^{ 2013} + 2013x ^{ 2010}$. I have tried finding the roots of $x$ from the given equation but that does not work.
3
votes
4answers
301 views

What is the shortest way to compute the last 3 digits of $17^{256}$?

What is the shortest way to compute the last 3 digits of $17^{256}$ ? My solution: \begin{align} 17^{256} &=289^{128} \\ &=(290 - 1)^{128}\\ &=\binom{128}{0}290^{128} - ... ...
1
vote
1answer
42 views

Negative powers in modular arithmetic

Suppose we have set $Z = \{0, 1, \dots, N-1\}$ with arithmetic operations modulo $N$; $a > 0$ is an element in $Z$. Is it possible that $a^{-1}$ does not exist but $a^{-n}$ exists for some $n$, $1 ...
6
votes
2answers
79 views

What are the number of integers a $1 \le a \le 100$ such that $a^a$ is a perfect square.

What are the number of integers a such that $1 \le a \le 100$ and $a^a$ is a perfect square. I think the answer should be 51 since a can be 1 and then 2,4,6,...100. Is the answer correct?
8
votes
5answers
158 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
2
votes
3answers
107 views

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ wrt $x$?

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ with respect to $x$?
6
votes
1answer
214 views

What does the raised $^2$ stand for?

What does the raised $2$ stand for? My first guess was: $4^2$ is $2\times 4=8$? Note: Am not really good at math
6
votes
2answers
1k views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
0
votes
4answers
76 views

Is the power of 1/2 same thing as principal square root?

$\sqrt{9} = 3$ 9 has 2 square roots: 3 and -3. What is $9^\frac12$? Is $9^\frac12 = \sqrt{9} = 3$ or is $9^\frac12 = \pm3$?
1
vote
1answer
46 views

Is there a formula for a sequence like $k^{t}-k^{t-1}+k^{t-2}-…+k^{2}-k^{1}+k^{0}$

I am trying to solve a programming problem and my intended solution involves a calculation like this one: $k^{t}-k^{t-1}+k^{t-2}-...+k^{2}-k^{1}+k^{0}$ The problem is that $t$ can be as large as ...
1
vote
1answer
51 views

Solving for exponent with multiple bases

From a practical perspective, my question can be most easily considered as solving for time in a future-value type equation, but for two separate investments growing at different rates. Say you have ...
0
votes
1answer
40 views

Pandigital exponent solution?

I don't have Reviewing C++ By Alex Maurea book. But someone in facebook post a question from this book which is following You can see the puzzle at page 542 problem 29. Now I think the answer ...
2
votes
4answers
188 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
2
votes
1answer
53 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
3
votes
4answers
107 views

Calculating $\log_7 125$

So the problem asks to calculate $\log_7 125$. It's multiple choice and the options are $2.48$ $4.75$ $1.77$ $2.09$ Given that $7^2 = 49$ and $7^3 = 343$, the answer must be either option 1 or 4, ...
7
votes
6answers
244 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
0
votes
1answer
48 views

Fractional exponentiation in modular arithmetic

Does raising a modular expression to a fraction mean anything? For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$. Does this violate the rules of modularity?
6
votes
2answers
121 views

What's the first digit of 2410^2410?

The first digit means the left most digit. 2410 is just an example and it can be replaced by any other numbers. Can any one help me to solve it?
-1
votes
1answer
55 views

How do I solve for $t$ and $s$ in $y = x^{-t/s}$?

I have $$y = x^{-t/s}$$ How do I solve for $t$ and $s$ in terms of the other variables?
1
vote
2answers
81 views

Why is $2^a > a^3$?

I found this rather interesting and maybe, a bit too obvious for some people property about 2 raised to some power. $2^a > a^3$, if $a=0,a=1 \text{ or } a\ge 10$ .($a \in N$) I seem to get a bit of ...
4
votes
4answers
135 views

negative exponent problem

$$\sqrt{\frac{1}{3^0 + 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}}$$ Does this equal = $$ \begin{align*} & \sqrt{3^0 + 3^1 + 3^2 + 3^3 + 3^4} \\ =&\sqrt{1 + 3 + 9 + 27 + 81} \\ =&\sqrt{121} \\ ...
1
vote
4answers
79 views

What exponent should I raise $26$ to in order to equal $2^{76}$?

I want to figure out how long an all-caps password needs to be to equal $2^{76}$ bits of security. I would type this into Wolfram Alpha, but I'm not sure what function to use or if it can compute ...