Questions about exponentiation

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1
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1answer
263 views

How to solve for the equation $ax \exp(bx)=c$?

How to solve for the equation $ax \exp(bx)=c$? It is known that $x\geq 0$.
2
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1answer
209 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
1
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2answers
86 views

Logarithmic equations and their related exponential equations

I am learning about logarithms and I'd like some examples of the following: Examples where the logarithmic value is a different positive integer Examples where the logarithmic value is a different ...
10
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4answers
531 views

Which step in this process allows me to erroneously conclude that $i = 1$

I was playing around with imaginary numbers and exponents and came up with this: $$ i = \sqrt{-1} $$ $$ \sqrt{-1} = (-1)^{1/2} $$ $$ (-1)^{1/2} = (-1)^{2/4} $$ $$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
17
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6answers
1k views

$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$

$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$
-1
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2answers
123 views

Proof of even like powers?

Can someone show me the proof that difference of like even powers of any two numbers is divisible by the sum of the bases?
0
votes
1answer
138 views

Calculating $f(x)$ using matrix exponentiation

Having polynomial in form $\sum \limits_{i=1}^{n} c_i x^i$ and linear recurrence, for example $$ R(n)=2 R(n-1)+3R(n-2) + f(n)$$ where $f(n)$ is our polynomial and $R(n)$ is linear reccurence, how ...
7
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1answer
2k views

Ways to calculate the derivative of the matrix exponential

Could someone provide me with a rigorous proof as to why the derivative of the function $f:t \ni \mathbb{R} \mapsto e^{tA}\in \textrm{Mat}_n (\mathbb{R})$ is $t \mapsto A\cdot e^{tA}$ ? I didn't ...
1
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2answers
740 views

Matrix exponential of a 2x2 matrix composed of a antihermitian matrix and a symmetric matrix

As per the title, I'd like to calculate the exponential of a matrix which has an antihermitian component and a symmetric component (although this fact may not be useful). More specifically ...
4
votes
3answers
149 views

Calculate $11^{35} \pmod{71}$

Calculate $11^{35} \pmod{71}$ I have: $= (11^5)^7 \pmod{71}$ $=23^7 \pmod{71}$ And I'm not really sure what to do from this point..
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
6
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0answers
192 views

The exponential map

I'm following a course about riemannian geometry, and I was fascinated with the exponential map. I was wondering what the reason of this name is... is there any relationship with the real and complex ...
5
votes
1answer
964 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
4
votes
3answers
818 views

Calculating $17^{14}\mod{71}$ using Fermat's little theorem

Calculate $17^{14} \pmod{71}$ By Fermat's little theorem: $17^{70} \equiv 1 \pmod{71}$ $17^{14} \equiv 17^{(70\cdot\frac{14}{70})}\pmod{71}$ And then I don't really know what to do from this point ...
4
votes
5answers
281 views

What is the remainder of $(14^{2010}+1) \div 6$?

What is the remainder of $(14^{2010}+1) \div 6$? Someone showed me a way to do this by finding a pattern, i.e.: $14^1\div6$ has remainder 2 $14^2\div6$ has remainder 4 $14^3\div6$ has remainder 2 ...
0
votes
0answers
121 views

Is it true that $n> a^2\Rightarrow n!>a^n$, $n\in\mathbb{N}, a\in\mathbb{R}$?

If so, how can it be proven? (I have evaluated it up to $n=25$.) If not, does there exist a $k\in\mathbb{R}$ such as that $n> a^k\Rightarrow n!>a^n$, with $n\in\mathbb{N},a\in\mathbb{R}$? It ...
2
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1answer
115 views

How to solve $x^x=k$? [duplicate]

Possible Duplicate: $x^x=y$. How to solve for $x$? If we have $x^x=4$ it's easily solved by substituting $x$ with $2$. But for general equation like $x^x = k$, how we can find the solution? ...
4
votes
2answers
109 views

Why does $16^{1/3} = 2^{4/3}$

I'm working through an example problem in my text book and they simplify $16^{1/3}$ to $2^{4/3}$. They also simplify $\frac{1}{2}(2^{8/3})$ to $2^{5/3}$. I don't follow the logic.
10
votes
10answers
324 views

Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
1
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2answers
1k views

Solving an equation in which the variable appears as both an exponent and a base

I am currently playing around with exponential and logarithmic functions, and am now trying to solve the following: $2^x = x^2$. My problem is that whenever I'm exponentiating or taking the logarithm ...
0
votes
1answer
79 views

Simplifying with negative exponents $(-11a^2)(-4a^{-7})$

$$(-11a^2)(-4a^{-7})$$ Can someone reformat, $a$ is second set of parenthesis is to the $-7$ power. Change to reciprocal so we get $$\left(\frac{1}{-4a}\right)^7 * \frac{11a}{1} = $$ confused ...
6
votes
3answers
439 views

How to solve $x\cdot\mathrm e^x=1$? [duplicate]

Possible Duplicate: Inverse of $y=xe^x$ I would like to solve the equation $x \cdot\mathrm e^x=1$. I know it has an answer, I could find it with a calculator, but I don't remember how to ...
9
votes
3answers
628 views

Who introduced the notation $x^2$?

In the book 'Problem Solving and Number Theory' I read The law of quadratic reciprocity was discovered for the first time, in a complex form, by L. Euler who published it in his paper ...
0
votes
1answer
114 views

How do I find p for equations of the form $\sum \limits_i \frac{a_i}{b_i^p} = 1$

The problem I'm facing is solving the following equation for $p$ given the constants $a_i$ and $b_i$: $$ \sum_i \frac{a_i}{b_i^p} = 1 $$ Is there a general technique that would allow me to find a ...
1
vote
1answer
100 views

compute an exponential complex number

I have a pretty basic question about complex numbers. If $z=x+yi$, a complex number, i want to compute the real and the imaginary part of the number $w=e^{e^z}$. Thanks in advance for any help.
8
votes
1answer
450 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
1
vote
0answers
221 views

What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
1
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2answers
61 views

Proof of a lower bound $\lambda(n)$ of the smallest number of multiplications $\ell(n)$ needed to compute $a^n$ for an integer $a$

Let $\ell(n)$ be the smallest number of multiplications needed to compute $a^n$ for any integer $a$. Here, a multiplication is $a_i := a_j \cdot a_k$ for $j, k < i$ and $a_0 := a$, e.g. $\ell(8) = ...
2
votes
0answers
86 views

Finding the value of $x$ for an equation

If we have the expression $a=x^{c\cdot x+1}$ where the values of $a,c$ are known, how can we find the value of $x$? I tried using log but it yields: $x = a ^ {(1/x)/(c-1/x)}$ from which I can't find ...
5
votes
1answer
458 views

Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
1
vote
3answers
319 views

A contradiction involving exponents

Where is the error in the following statement: $i^2=(i^2)^{\frac{4}{4}}=(i^4)^{\frac{2}{4}}=(1)^{\frac{1}{2}}=1$? I feel the error is in the first equality, because $(i^2)^{\frac{4}{4}}$ is in fact ...
10
votes
5answers
405 views

Why is the math for negative exponents so?

This is what we are taught: $$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$ but I don't understand why we take the inverse of the base when we have a negative exponent. Can anyone explain why?
4
votes
3answers
370 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
15
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5answers
2k views

Approximation of $e^{-x}$

Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
5
votes
3answers
376 views

Proof of exponential from homomorphism property

I am going back through a bunch of calculus I learned in high school and proving the stuff that they just told us was true. Along the way, I found I had to prove that if $f(x+y)=f(x)f(y)$ then $f$ is ...
2
votes
2answers
357 views

What is the rule of equating exponents called?

For example: $$2^{2n-1} = 2^{n+2} \Rightarrow 2n - 1 - n - 2 = 0 \Rightarrow n = 3$$ I couldn't find this rule in properties of exponents i.e when the bases are equal, the exponents can be equated. ...
0
votes
1answer
62 views

Which loan type is cheapest?

I have a 4% loan that spans 20 years, where I pay a fixed amount every three months. If I make an extra payment, I then can choose between two options keep the duration of the loan constant, and I ...
3
votes
2answers
482 views

Solving tricky Knuth Up Arrow Notations

How would I solve something like $2\uparrow\uparrow n$? when n ≤1? Or $2\uparrow^{-2}2$? Thanks!
0
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1answer
1k views

Can a variable appear as a denominator in a linear equation or is the system non-linear?

I am trying to figure out whether the following equation is non-linear or if it's linear, how would I solve it? $x+\frac{2}{y}=0$ It can be rewritten as $x+y^{-2}=0$ so I guess if this is ...
0
votes
1answer
69 views

Re-writing a logarithm to a power

Given: $$(4\ln x)^2$$ Is this simplified to $8\ln x$, (multiplying the expression by 2), $32\ln x$, (square $4$ ($16$), then $\ln x$ ($2\ln x$) and combine again), or something else? Just to be ...
1
vote
1answer
111 views

Branching of $z^z$

I'm facing the problem of proving that ($0,+\infty)$ is a branching point of $f(z)=z^z$. Now, straight from the definition, if $z=|z|e^{i\theta}$ with $0 \le \theta < 2\pi$ $$ z^z := e^{z\log z} = ...
9
votes
4answers
426 views

Intuitive explanation of $(a^b)^c = a^{bc}$

What is an intuitive explanation for the rule that $(a^b)^c = a^{bc}$. I'm trying to wrap my head around it, but I can't really do it.
13
votes
2answers
1k views

How to know if a number is a power of $x$

I couldn't find anything on the Internet which could direct me to the solution of the following problem. I want to know if $n$ can be calculated by $x^y$ where $y\ge 2$ and $x\ge 2$. I tried using ...
0
votes
1answer
57 views

How to express this exponential equation in terms of $m$?

How can I express the equation $c = 2^{m+1} - 2^m$ in terms of m? t.i.a.
2
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1answer
105 views

Is it possible to know if sums of powers of a number is divisible by another number?

Is there a way to find whether a number (say $A$) formed by summing powers of another number (say $B$) is divisible by another number $C$? $A$ is a number like, for example, $B^1+B^3$. We can use a ...
5
votes
2answers
2k views

How can I calculate non-integer exponents?

I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function: $$ f(x,y) = \begin {cases} 1 & y = 0 \\ (x)f(x, y-1) & y > 0 \end ...
7
votes
5answers
433 views

Proof that $6^n$ always has a last digit of $6$

Without being proficient in math at all, I have figured out, by looking at series of numbers, that $6$ in the $n$-th power always seems to end with the digit $6$. Anyone here willing to link me to a ...
3
votes
4answers
199 views

What's the easiest way to find a decreasing exponential function passing through two given points?

What's the easiest way to find a decreasing exponential function passing through two given points? $(x_1,y_1)$ and $(x_2,y_2)$ are given. All coordinates are $\ >\ 3$.
5
votes
2answers
361 views

How many positive integer solutions to $a^x+b^x+c^x=abc$?

How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to ...
3
votes
1answer
644 views

Integral of exponential function with trigonometric identities

I need help in solving the following definite integral. I could not find any example like this $$\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; ...