Questions about exponentiation

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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 $80$?)
5
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1answer
147 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
25 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
26 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
22 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
54 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 ...
4
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2answers
115 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
22 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
31 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
41 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 ...
2
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3answers
161 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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3answers
141 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
30 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
42 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
41 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
40 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
37 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
38 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
37 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
61 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
370 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
92 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
83 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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0answers
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 ...
0
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3answers
43 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
40 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
48 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= ...
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1answer
30 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
0
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1answer
42 views

How can I derive this summation?

I have the following equation, $$ K_r=\left ( \frac{P}{RT} \right )^{v}exp \left \{ \sum_{s}\left [ (\beta_{s,r}-\alpha_{s,r}) \left \langle \frac{h_s}{RT}-\frac{s_s}{R}\right \rangle \right ] ...
0
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2answers
49 views

Convergence of a series ${}\qquad{}$

Does this series converge? $$\sum_{x=2}^n \left(\frac{1}{x}\right)^{\left(\frac{1}{x}\right)}$$ I tried hard but stil had problems... Could someone help me?
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1answer
26 views

Could somebody validate my proof for the limit of $a^{x_n}$ when $x_n \to c$?

So, here is the clear formulation of the problem: let $(x_n) $ be a convergent sequence of positive numbers, with $x_n \to c$. I want to prove that the sequence $(y_n) $, with $y_n=a^{x_n} $, ...
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0answers
17 views

How we can calculate the power of an interval?

We know if two intervals are uncorrelated like $X=[a,b], \; Y=[c,d]$ the product of $X$ and $Y$ is: $X \times Y = [\min(S),\max(S)], \; S = (ac,ad,bc,bd)$ But for powers, if the intervals are ...
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2answers
189 views

Properties of Logarithms

How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms? ...
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0answers
117 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
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0answers
33 views

How to calculate sum of digit of a power

Find the sum of the digits of: $$\left\lfloor\frac{k^{h+1}-1}{h-1}\right\rfloor$$ I need to calculate sum of digits in answer. Note as $k$ and $h$ can be a very big value, answer is getting ...
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1answer
26 views

Economics question. Rehashing the basics of dealing with exponents

Struggling to put two equations together effectively again. I have my income equation: $$ y=zk^\alpha $$ And I'm trying to plug it into my marginal product of capital $$ MPK=\alpha z k^{\alpha-1} ...
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4answers
82 views

What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
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2answers
36 views

Simplifying $\exp {- i 2 \pi / N}$.

a and b are complex numbers and I know the equation below. $$X_{N} = a + e^{-i2\pi /N}*b$$ I wanted to simplify it. Here is what I've tried. I know $e^{-i\pi} = -1$ $X_{N} = a + \left ( e^{i\pi} ...
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2answers
23 views

Exponents with variables inside exponents

I am confused about how to reduce this, is there any way? $\sqrt x \ln(\sqrt x) = ln(\sqrt x^\sqrt x) = $ $?$ This can be written like this too: $ln(x^\frac{x^\frac{1}{2}}{2}$) Or: ...
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0answers
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Find the value of x if $\log_{2}({5 * 2^x + 1})$, $\log_{2}({2^{1 -x} + 1})$ and 1 are in arithmetic progression

My try: $1 - \log_{2}({2^{1 -x} + 1})$ = $ \log_{2}({2^{1 -x} + 1}) - \log_{2}({5 * 2^x + 1})$ $10 * 2^x + 2 = (2^{1 -x} + 1 )^2$ Let $2^x = t $ $10t^3 + t^2 -4t -4 = 0$ But I ...
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2answers
26 views

Non-integer exponents of negative numbers?

There is a formula for exponents of negative numbers as follows: $m^n=(-1)^n|m|^n$. This formulation works when $m<0$ and $n\in \mathbb{Z}$. But what about for $n\in \mathbb{R}$? Is there a ...
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2answers
24 views

no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$

The given equation is $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$ I took $\log_{2}{x}$ = $t$ and then rewrote the given equation as $$x^{3t^2 + 4t - 5} = \sqrt{2}$$ ...
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3answers
243 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
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1answer
48 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
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2answers
47 views

No real solution to logarithmic equation?

$$e^x + 1 = 2e^{-x}$$ Wolfram Alpha claims no real solution and my text book claims the solution $x=0$. Why can't I simply multiply each side by $e^x$: $$e^{2x} + e^x = 2$$ $$\ln(e^{2x}) + ...
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1answer
44 views

If $\log_{30}{3} = c$ and $\log_{30}{5} = d$ then the value of $\log_{30}{8} $ is??

I attempted the following: $\log_{30}{8} = 3\log_{30}{2}$ $\log_{30}{3} = c$ is equivalent to $3 = 30^c$ $\log_{30}{5} = d$ is equivalent to $5 = 30^d$ What should I do further?? Is ...