Questions about exponentiation

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1answer
68 views

Trying to convert a nasty logarithm into an exponential

I have the following equation that I must express in terms of $r$: $$\Delta V = \frac{\lambda}{2 \pi \epsilon_0} \ln(\frac{r}{R})$$ This is a pretty tough one. I am not sure how to get the r out of ...
5
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1answer
419 views

Prime decomposition of an integer: methods of determining the prime factors $ p_1, p_2, …, p_r$ and powers $k_1,k_2, …, k_r$

Any integer n can be written in the form $ n = p_1^{k_1}p_2^{k_2} ... p_r^{k_r} $, where the powers $ k_1, k_2, ...,k_r $ are integers and $ p_1, p_2, ..., p_r$ are primes. Now I am interested in ...
3
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1answer
81 views

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 $f(t)=ae^{kt}$?
3
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3answers
113 views

How can we represent any number as a series of exponents?

Say I have a positive integer that is one-thousand digits long. What math could I use to represent this number as a series of exponents in a significantly shorter form than the original number? The ...
2
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3answers
289 views

Calculating powers

I was thinking how I could program powers into my application. And I want to do this without using the default math libraries. I want to use only +,-,* and /. So I wondered what is the definition of a ...
0
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1answer
164 views

Modular arithmetic: mod p-1 after exponentiation?

I keep coming across proofs that seem to use the following derivation, but I'm unsure where it comes from. What theorem shows that this is a correct step to take? $ g^{x} = g^y$ mod $p$ $\iff$ $x = ...
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5answers
198 views

Derivative of $f(x)^{g(x)}$

Question: Find the derivative of $$ f(x) = \left(\frac{1}{x}\right)^{\Large\frac{1}{x}} $$ Tip: convert $f(x)$ to $e^{g(x)}$. How does one convert $f(x)$ to $e^{g(x)}$? Thanks in advance
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5answers
237 views

Solving an equation with a logarithm in the exponent

I try to solve the following equation: $$ (N+1)^{\log_N{125}} = 216 $$ I know the answer is 5 here but how could I rewrite the equations so I can solve it? I tried to take the log of both sides but ...
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2answers
106 views

How to efficiently compute the coefficients in a bi-binomial expansion?

Is there a computationally efficient way of calculating the coefficients of the polynomial expansion of expressions like $(1+x^a)^m(1+x^b)^n$ for arbitrary positive integers $m,n,a,b$ (and especially ...
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1answer
42 views

Logarithm exponent in Chernoff bound

I am applying Chernoff bound for a Poisson process with mean $\lg n$. I am putting $\delta =4$. Hence, $Pr(X<(1+4)\mu)< (\frac{e^\delta}{(1+4)^{(1+4)}})^\mu$ $ = (\frac{e^\delta}{5^5})^{\lg ...
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2answers
153 views

pow$(X,Y)$ $>$ pow$(Y,X)$, if $X<Y$.

How can we proof following? if $X < Y$, then: $X^{Y} > Y^{X}$ , Where X, and Y are integers. Also $X,Y > 1$. Except a special case $2^{3} < 3^{2}$. I think for other ...
0
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1answer
224 views

Comparing two exponentiations

$b^n$ where the base $b$ is a positive integer greater than $1$ and the exponent $n$ is a rational number in simplified form. How would one compare (resulting in <, =, or >) two such ...
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2answers
354 views

Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime

My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
5
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3answers
228 views

Laws of Exponents for Cosets - Fraleigh p. 142 14.22

A torsion group is a group all of whose elements have finite order. A group is torsion free if the identity is the only element of finite order. Prove: if G is a torsion group, then so is G/H ...
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3answers
386 views

Proof for $\frac{x^n}{x^n}=x^{n-n}$

Does the equality $\forall n,x\not=0: \frac{x^n}{x^n}=x^{n-n}$ come straight from the definition of exponents, or is a more elaborate proof needed? Please note that I'm not asking about $x^0=1$, but ...
5
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4answers
278 views

Can we prove $a^{\log_bn} = n^{\log_ba}$?

Can we prove $$a^{\log_bn} = n^{\log_ba}?$$ I forget how to prove this theorem. I picked up one numbers for test, and they worked.
3
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3answers
138 views

$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$

$$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$$ Is that correct? I would've done $$ 2^{10 - x} \cdot 2^{10 - x}\;\; = \;\; (2)^{10 - x + 10 - x} \; = \; (2)^{2 \cdot (10 - x)} \;=\; 4^{10 - x}\tag{1} $$ ...
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2answers
99 views

Understanding $\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$

I'm going through a book about algorithms and I encounter this. $$\frac {b^{n+1}-a^{n+1}}{b-a} = \sum_{i=0}^{n}a^ib^{n-i}$$ How is this equation formed? If a theorem has been applied, what theorem ...
8
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1answer
151 views

Proving a number defined by a sequence is a square number

I found this problem in a math magazine: Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by: $$ x_0 = 0\\ x_1 = 1\\ x_{n+2}+x_{n+1}+2x_{n}=0 $$ Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
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10answers
5k views

How to solve $x^{1/2}-x^{1/3} = 0$

How can I solve the following equation? I really can't figure out how to solve it: $x^{1/2}-x^{1/3} = 0$ Thank you.
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1answer
18 views

How to find an algebraic term when 2 similar exponential forms of it are given?

if $a^2bc^3 = 25$ and $ab^2= 5$, how would you find $abc$? Is there a formula of which these are a part of? What I've done is this: $$abc^3 = \frac{25}a\quad\implies\quad abc = ...
6
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1answer
510 views

System of Equations With Exponents

What are the steps to solving a system of equations when $x$ and $y$ are exponents? Here is the problem to solve for $x$ and $y$. $$8^{3x}=4^{2y}$$ $$x-y=5$$
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5answers
128 views

Solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$

What is the right solution for $\log_7x+\log_{\frac17}x^2=\log_{49}x-3$. What logarithm identities used?
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0answers
77 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...
2
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2answers
82 views

Proving that $x^a=x^{a\,\bmod\,{\phi(m)}} \pmod m$

i want to prove $x^a \equiv x^{a\,\bmod\,8} \pmod{15}$.....(1) my logic: here, since $\mathrm{gcd}(x,15)=1$, and $15$ has prime factors $3$ and $5$ (given) we can apply Euler's theorem. we know that ...
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1answer
117 views
0
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2answers
80 views

What are these numbers called?

Say I have numbers that are all multiple of 2, I would say, well they are multiples of two. How are numbers $x$ called with respect to $a$ that are all formed like $x = a^b$? I am assuming here that ...
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4answers
689 views

Value of $k$ for which $e^x = kx$ has $1$ solution

I need to work out the value of $k$, where $k>0$, for which $e^x=kx$ has $1$ solution. I've done it somewhat intuitively as follows: $e^x=kx$ By inspection we can see that when $x=1$, the ...
6
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3answers
188 views

What's the intuition behind non-integer exponents/powers

Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. Is there some intuition to be had for the number $a^x$? For example the intuition of $a^2$ is obvious; it's $a*a$ which ...
4
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3answers
171 views

What is the difference between exponentials and powers?

I am a java programmer. But I have a doubt regarding a mathematics. There was a method called Math.exp(double a) description:Returns Euler's number e raised to the power of a double value. and another ...
2
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5answers
279 views

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
4
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1answer
96 views

What's the arity of the factorial and exponential operations?

I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the ...
5
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0answers
221 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
4
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1answer
149 views

Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
1
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1answer
147 views

Solve for variable inside multiple power in terms of the powers.

I'm a programmer working to write test software. Currently estimates the values it needs with by testing with a brute force algorithm. I'm trying to improve the math behind the software so that I can ...
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0answers
43 views

can´t solve this kind of exponential equations

I have no clue how can i solve this kind of exponential equation in closed form: $a^n - b^n \le c$ where $a > 1$ and $-1 < b < 0$ thank you very much for your help
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2answers
73 views

Computing $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$

I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$. But honestly I'm not even sure where to start. ...
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0answers
197 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
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5answers
472 views

Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$

I'm so puzzled about this: $$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$ Why isn't $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that ...
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0answers
173 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
11
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4answers
727 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
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18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
4
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1answer
291 views

Summing 2 to the power of the subset sums of a power set

Sorry in advance, as I suspect I lack both the proper terms and the proper notation for the problem I have, but I'll try to be clear. If I have a set $S = \{1,2,3\}$, I figured out that the summation ...
3
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5answers
172 views

Summation over exponent $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$

Why does $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$, where does that 3 comes from? Ok, from your answers I looked it up on wikipedia Geometric Progression, but to derive the formula it says to multiply by ...
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7answers
593 views

Numbers to the Power of Zero

I have been a witness to many a discussion about numbers to the power of zero, but I have never really been sold on any claims or explanations. This is a three part question, the parts are as ...
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0answers
91 views

Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
0
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1answer
107 views

How do I solve this exponential equation?

$$x = 2^{x-3}$$ Does there exist an analytical solution to this equation? If so, how do I find it? What if it is changed to an equality? $$x>2^{x-3}$$
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4answers
923 views

Integration with infinity and exponential

How is $$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$ however my answer comes zero because putting limit in the expression, we get: $$\frac1\infty\left(-\frac1{2a}\right) ...
0
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1answer
116 views

Solution for an exponential expression without using logarithms, with two defined variables

If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$? This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just ...
3
votes
1answer
156 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...