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

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2
votes
2answers
50 views

$2^n$ modulo n where n is odd always yields either an even or $1$

I'm attempting to do a pidgeonhole proof to prove that for some odd integer n, there is always a $2^k$ such that $2^k \mod(n) = 1$. I know that $2^n \mod(n)$ will always yield either an even number or ...
1
vote
2answers
26 views

Continuous compounding question

A population of rabbits starts out with $100$ rabbits. The growth rate is $11.7$% per day. Determine the exponential equation. Is it $$\mathbb {P(t)} = 100e^{11.7t}$$ Can you guys give me the ...
0
votes
2answers
128 views

Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$

Hi guys this was a practice problem I was given, can anyone help me out on it? This is the problem: Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$ and the following is what I ...
-1
votes
4answers
148 views

Does $\infty^0=1$?

I was wondering if $\infty^0=1$. Some people have told me that there is no answer; it is undefined. Others have told me that the answer is $1$, using the rule $a^0=1, \ a\neq 0$. If it is truly ...
0
votes
1answer
58 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
1
vote
1answer
127 views

Rising/Falling Powers, Summation

1) Show that $$(-n)^{\bar p} = (-1)^n n^\underline{p}$$ (original screenshot) 2) Evaluate the sum $$\sum_{a\le n\lt b}n^{\bar p}$$ (original screenshot) Thoughts regarding question 1: I've ...
3
votes
2answers
145 views

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 ...
2
votes
2answers
778 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
0
votes
2answers
151 views

Why is X raised to the power of 0 = 1?

So as the topic states, why is 5^0 = 1 and not 5 or 0? Is it only because the other exponential laws wouldn't work if it was not the case?
-1
votes
1answer
44 views

How to solve the integral $\int x\cdot 9\cdot x^{2x^2} dx$

How to solve the integral $$\int x\cdot 9\cdot x^{2x^2} dx$$ I tried $u=2x^2$ and $du= 4x\;dx\Longrightarrow$ $$\int x\cdot9\cdot x^{2x^2}\;dx=\frac94\int x^u du$$But it was unable to pass.
3
votes
4answers
195 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 ...
0
votes
2answers
74 views

Converting powers of 3 into powers of 2? [closed]

I'm stuck on a problem. I have a term $3^m$ where $m$ is an integer $> 0$. and I want to represent it as $2^m + a$ however I don't want to keep the $a$. I am looking for a formula to represent $a$ ...
2
votes
2answers
88 views

How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...
0
votes
1answer
32 views

About definition of $a^b$ with $a \in \Bbb{R} \wedge b \in \Bbb{N}$

-- let $a,c \in \mathbb{R}$, and $b \in \Bbb{N}$, with $\Bbb{R}$ is a complete ordered field, $c \triangleq a^b$ if $c=\begin{cases} 1, & \mbox{if } a\neq 0 \wedge b=0\\ 0, & \mbox{if } a=0 ...
2
votes
3answers
94 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
2
votes
0answers
53 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
0
votes
3answers
52 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
0
votes
1answer
143 views

How do I solve $|\sinh(x+iy)|^2 = (\sin(y))^2+(\sinh(x))^2$

How do I solve this ? $|\sinh (x+iy)|^2 = ( \sin (y))^2+ ( \sinh (x))^2$ I'm not sure how to solve the left hand side.
5
votes
6answers
256 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
4
votes
2answers
118 views

Is it possible to solve $i^2+i+1\equiv 0\pmod{2^p-1}$ in general?

While looking at the Mersenne numbers (for prime $p$, the number $2^p-1$), I noticed that only certain of them had any solution to the modular equation $i^2+i+1\equiv 0\pmod{2^p-1}$, e.g., ...
1
vote
2answers
60 views

modular exponentiation where exponent is 1 (mod m)

Suppose I know that $ax + by \equiv 1 \pmod{m}$, why would then, for any $0<s<m$ it would hold that $s^{ax} s^{by} \equiv s^{ax+by} \equiv s \pmod{m}$? I do not understand the last step here. ...
3
votes
1answer
67 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 ...
0
votes
3answers
68 views

Logarithm properties doubt

The problem is $\log (5.64)^4$. According to the properties and laws of exponents, $\log (m^r) = r \log (m)$. But since the exponent is outside of the parenthesis in this problem, does it solves by ...
4
votes
1answer
199 views

Is $n^{\log c} = c^{\log n}$ true?

Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.
2
votes
2answers
481 views

How to find out the value of numbers having fractional powers

How to find out the value of numbers having fractional powers manually without using logarithms and calculators?? For example : $2^{1.6}, 3^{2.1}, 5^{3.22}$ etc, I know we can find out the value using ...
0
votes
1answer
37 views

Is there a seperate something in front logarithm that is raising a base to a power?

I am trying to solve a problem with the following form $$e^{\displaystyle A\log(x)}$$ $e^{\log(x)}$ is simply $x$, but how do I go about separating the $A$?
2
votes
2answers
697 views

x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
1
vote
1answer
30 views

Proof for exponentiation in modular arithemetic

I have found out, that the following is true for modular arithmetic when $t$ is a natural number. $$a^t \bmod\ n \equiv (a\bmod\ n)^t\bmod\ n$$ But I have been unable to find a proof for this, does ...
10
votes
1answer
217 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 $n\times n$ ...
3
votes
2answers
87 views

Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective?

This is related to a personal exploration of isometries of directed graphs, motivated by my son's Lego Duplo train tracks and identifying "interesting" layouts. If $M$ is the adjacency matrix for a ...
0
votes
4answers
73 views

How to compute the exponent?

So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$? Thanks in advance!
2
votes
2answers
92 views

Solution for $4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$

Im trying to get a solution for: $4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$ My main problem is thati dont now how to combine this potencys! Ive also thought about another function that would bring me ...
1
vote
3answers
74 views

Explanation for limits equality.

$$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{1}{x}}} = \exp \left( {\mathop {\lim }\limits_{x \to 0} \frac{{\frac{{{a^x} + {b^x}}}{2} - 1}}{x}} \right)$$ I am ...
1
vote
0answers
40 views

Algebra of Exponential and Log Functions

This question may have a simple answer or a very complex one, but I am interested in what the reasons are for logarithms and exponential functions having the properties they have. To my knowledge ...
0
votes
1answer
43 views

Proof of generalization of a particular limit converging to $e^{\frac{1}{(p-1)^2}}$

I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th ...
0
votes
1answer
54 views

simplification of squared expression

I haven't done a math class in a while and I'm stumped on what seems to be a simple question. How does $$ ( [(k + 1)^2]/4 ) (k + 2)^2$$ simplify to $$ [ (k + 1)(k + 2)/2 ]^2 $$ What rule of ...
1
vote
1answer
38 views

Relating to $( a ^ b ) + c = d$ and $( a ^ b ) - c = d$

Is there a way of deducing the smallest integer values for $a, b$ and $c$ that satisfy either $( a ^ b ) + c = d$ or $( a ^ b ) - c = d$ such that the addition $( a + b + c )$ is the smallest ...
0
votes
1answer
31 views

Exponential equivalent for geometric space

I'm just starting a foray into geometric algebra and calculus so that I can develop a geometric version of the standard arithmetic neural net. Specifically when calculating the error function for a ...
0
votes
1answer
100 views

Will negative bases with irrational exponents get a real or imaginary number?

Here are a few examples: $$(-1)^{\sqrt{2}},(-2)^{\pi},(-3)^{e}$$ From what I've learned, negative bases must have denominators of the exponent odd. Normally if we do $(-2)^{0.258}$ it would be the ...
2
votes
3answers
84 views

Why can matrix exponentiation be done by squaring?

Matrix multiplication is not communative: A*B != B*A Then why can matrix exponentiation be done by squaring? I have tried searching for special cases where this rule did not apply, but from what I've ...
1
vote
2answers
72 views

Why the identity $ a^x = a^y \Longrightarrow x=y $ do not work for $a<0$?

All books that i am reading are telling that the identiy $ a^x = a^y \Longrightarrow x=y ; \,\,\,\,\, a \in \mathbb{R} - \{0,1\} $ $ \,\,\,\,$ also do not work for $a<0$, but, for example, if $ ...
-1
votes
1answer
20 views

exponentiation with the base between 0 and 1

i have a variable $c$ such that $0\leq c\leq1$ and a variable $t>0$ can i always say that $0\leq(c^t)\leq1$ ? What i found difficult is the exponentiation between for example $0<c<1$ and ...
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
137 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
123 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
122 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
262 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, ...