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

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negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
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2answers
54 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
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1answer
65 views

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as ...
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1answer
47 views

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. ...
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5answers
84 views

Factorial of zero is 1. Why? [duplicate]

Why is the factorial of zero, one. What is the mathematical proof behind it?
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0answers
77 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
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1answer
34 views

Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
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3answers
43 views

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$ My try: $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$ $log_e{(x-2y)^2} = log_e{xy}$ ...
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3answers
51 views

$M$ matrix, $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ iff $M$ is not diagonalizable

M is a $n\times n$ matrix over $\mathbb R$. with $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ if and only if $M$ is not diagonalizable. I really don't know how to start thinking about this.. :/ I'd ...
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1answer
21 views

Proving increasing function, base < 1, exponent increasing

For a fair lottery game where the odds of $1$ ticket winning are $1$ in $p$, where you can spend a total of $K$ dollars, and where you will spread your ticket purchases equally among $n$ draws, prove ...
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1answer
23 views

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 ...
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0answers
11 views

An extension of the Golden-Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality is correct, at least in some cases: $$tr\left(A e^{B+C} \right) \leq tr\left(A ...
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2answers
75 views

Simplifying $x^i$ to real numbers

I have been studying power functions, and started to think about imaginary powers. Take the function $x^i$. Because I don't know how to multiply a number $i$ times, I tried to simplify the equation ...
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0answers
24 views

Solving sum of one variable with real exponents

I'm working with an annoying maximisation problem at the moment. I've spent a long time Googling, but I'm not having much success and I suspect it would be simple enough if I had the right tools. I ...
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1answer
29 views

Calculus / exponential

Find values of a and b so that $y = a·b^x$ and the line $y = x + 2$ are tangent at $x = 0$. I tried to substitute with the zero and it seem that the $B=1$ at all time but what about the $a$?
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2answers
49 views

Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

How to write: $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in $$(a+i b) $$ $$ ?$$ I tried to multiplicate by $$e^{i}$$ (the numerator and ...
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1answer
46 views

Infinite Perfect power of numbers in a certain form

A question I found very interesting , which I found written on a blackboard while visiting a near by community science center is as follows. Prove that there exist infinitely many $m,n,k$ for ...
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1answer
35 views

Lower bound for $(x + y)^k $?

I'm wondering, is there a lower bound for $(x + y)^k $? For example, if $x,y,k > 0$, can we say that $(x + y)^k \geq x^k + y^k$? If anyone has a source/reference for this, that would be great.
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1answer
37 views

Finding an exponential formula passed upon the start and end points.

I'd like to create pricing curve that's based upon a reverse exponential function. I know the starting point and ending point, but don't know how to create the curve in between. For example, say for ...
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4answers
74 views

If $10^{80}=2^x$, what is the value of $x$?

If $10^{80}=2^x$, what is the value of $x$? (Or, what binary word length would you need to contain $10$ to the $80$?)
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1answer
149 views

Solving the equation $a ^ b + b ^ a = 200$

Find $a$ and $b$, $a ^ b + b ^ a = 200$ One of the answers is $a = 1$ and $b = 199$. Lets say $a, b$ belongs to $\mathbb{R}$ then there will be many solutions, for each $a$ there exist $b$, in ...
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1answer
30 views

Find highest power of 2 that divides $3^{2^k}-1$

I am trying to find highest power of 2 that divides $3^{2^k}-1$ but I have no idea where to start - could you give me any hint?
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2answers
28 views

Exponent of Projection (T^2=T)

T:V->V is a linear transformation, also it's a projection, i.e. T^2=T. Find e^T. I thought of using the fact that if T=T^2 then e^T=e^(T^2) but I guess that doesn't work because exponent is a sum of ...
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1answer
23 views

Exponent of polynomials (of matrices)

$A$ is a matrix over $\mathbb R$ (reals). Prove that for every $f,g\in \mathbb R[x]$, $\displaystyle e^{f(A)}\times e^{g(A)} = e^{f(A)+g(A)}$ I tried using the sigma writing but got stuck (I ...
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1answer
56 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
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2answers
124 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
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1answer
32 views

Fractional index fallacy

According to index laws, $(a^b)^c=a^{b\cdot c}=(a^c)^b$ However, if for example we have $a=-1, b=4, c=1/2$, then we get the equation: $$((-1)^4)^{1/2}=(-1)^2=((-1)^{1/2})^4$$ The first equation is ...
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1answer
69 views

Is there any easy way to find the positive integer solutions $(x,y,z)$ from this linear equation?

The equation is like this: $3^x -2^y = 19^z$ It seems that no way to find the solution except using trial and error. I got only one solution: $x=3, y=3, and z= 1$ by using trial and error. But, ...
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0answers
24 views

Proof of equations of exponentials

I am working on a proof of exponential equations. Before this, I have verified the correctness, or approximate correctness of the equations by numerical results. The questions is: Given: 1) ...
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3answers
66 views

How to prove that $\frac{a^n}{a^m}$ is equal to $a^{n-m}$? [closed]

How to prove that $\dfrac{a^n}{a^m}$ is equal to $a^{n-m}$? Thank you in advance.
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1answer
35 views

Dividing 1 by powers of ten

I have something like, say, 3.9 x 10^-22 and I want it to be divided by 1. This is common in Chemistry science questions, but I fail to get the thing. How can we divide 1 by 3.9 x 10^-22 and get a ...
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1answer
44 views

Reference Request: Nicole Oresme history

It says on Wikipedia that [Nicole Oresme] also worked on fractional powers, and the notion of probability over infinite sequences, ideas which would not be further developed for the next three and ...
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3answers
163 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
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144 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
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1answer
33 views

How to simplify this expression that contains exponential terms?

In a multiple choice exam , I encountered the following question. The answer to the question is $$ \frac{17}{8}.$$ The question is: $$\frac{16^{x+1}+4^{2x}}{2^{x-3}8^{x+2}} \text{ is ? }$$
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1answer
43 views

Exponential function to prove [closed]

how would you prove that $Ae^x+Be^{-x}=A \sinh x+B\cosh x$ Thank you.
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1answer
46 views

Describing the exponantiation of a number by itself

When a mathematician says What is the square of $n$? It is generally understood that the expected answer would be to multiply $n$ by itself, $n^2$. Is there a word analogous to square to ...
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1answer
48 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
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1answer
38 views

How fast will a shape grow if it can grow exponentially only at the border, and growth is limited by crowding?

Take a hypothetical bacterium which divide once per minute. After $n$ minutes there will be $2^n$ bacteria, assuming no constraints. But what if its growth is constrained by resources and space? I am ...
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0answers
88 views

Finding of Exp(Jt), there J - Jordan form

I need to solve this: x(t)=exp(At)x0 To find exp(At) i need find exp(Jt), J(A) - Jordan form of A; ...
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0answers
42 views

Quick question on matrix exponentiation.

Hi can someone explain a step in the proof that if A, B are commutative matrices then $e^Ae^B=e^{A+B}$ So define $f(t)=e^{tA}e^{tB}$ then $f(0)=I$ and we have that $f'(t)=Ae^{tA}e^{tB}+e^{tA}Be^{tB}$ ...
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1answer
31 views

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 ...
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1answer
62 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
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3answers
381 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
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2answers
98 views

Is there a way to write $2^3+2^2+2^1+2^0$ in short form or a better way?

I am doing a question and instead of going through phases solving the question I was wondering if I could do it all in one with a short equation. The question is about compound interest finding the ...
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1answer
162 views

Proof that irrational exponents exist, and are unique?

Can someone provide a proof that for any given irrational number, $b$, exponentiation by that number defined as a limit of rational powers always converges, and that if we choose a particular base, ...
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26 views

Imaginary Exponentiation, and Exponentiation in General [duplicate]

I think I may have asked a similar question before, but first-how does one go about computing imaginary exponentials? Secondly, what is the formal definition of an irrational (or I guess more ...
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3answers
44 views

Is there a way to find x when x is an exponent?

I'm kind of stuck on how to solve the following. $10^x = 5^9$ Is there a method or a simple trick to find what is x?
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1answer
46 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
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1answer
49 views

Power Factoring Contest Question

The question was as follows: Compute the smallest positive integer $n$ such that $n^n$ has at least $1,000,000$ positive divisors. I did some work, finding that if $n=2^a*3^b*5^c*7^d$ then the $n^n= ...