Questions about exponentiation
15
votes
7answers
1k 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.
0
votes
1answer
16 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:
$$abc^3 = \frac{25}a\quad\implies\quad abc = ...
6
votes
1answer
255 views
System of Equations With Exponents
What are the steps to solving a system of equations when $x$ and $y$ are exponents? Here is the problem to solve for $x$ and $y$.
$$8^{3x}=4^{2y}$$
$$x-y=5$$
1
vote
5answers
105 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?
1
vote
0answers
41 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
votes
2answers
49 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 ...
1
vote
1answer
74 views
0
votes
2answers
49 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 ...
1
vote
4answers
168 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
votes
3answers
148 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 ...
3
votes
3answers
95 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
votes
6answers
244 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
votes
1answer
85 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
votes
0answers
102 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
votes
1answer
114 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 ...
0
votes
0answers
37 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
0
votes
2answers
47 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. ...
1
vote
0answers
75 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) = ...
3
votes
5answers
133 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 ...
2
votes
5answers
426 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 ...
0
votes
0answers
29 views
Solve for a variable to the power of itself [duplicate]
Possible Duplicate:
Is $x^x=y$ solvable for $x$?
I have stumbled upon a seemingly clear equation, which proved impossible to solve algebraically.
Consider $n^n = 2^t$. Is there a way to ...
1
vote
0answers
59 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
votes
1answer
80 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?
$$x>2^{x-3}$$
1
vote
4answers
264 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:
$$\frac1\infty\left(-\frac1{2a}\right) ...
0
votes
1answer
47 views
Solution for an exponential expression without using logarithms, with two defined variables
If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$?
This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just ...
1
vote
2answers
224 views
Solving system of recurrence relations
Base Case:
$$
\left\{
\begin{array}{c}
T(1) = 1 \\ T(2) = 1 \\T(3) = 4\end{array}
\right.
$$
I have the system:
$$
\left\{
\begin{array}{c}
T(N) = G(N-1) + F(N-1) \\
G(N) = F(N-1) + G(N-1) \\
...
5
votes
7answers
444 views
Is $0^0=1$ postulate independent of all other axioms of complex numbers?
This question is inspired by the other question which asked for a proof that $i^i$ is a real number.
Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
2
votes
1answer
194 views
Why use radical notation instead of rational exponents?
I'm helping my younger sister for her math class. She has recently been taught integer exponents, and has starteed studying radicals (mainly square roots). The next topic will be rational exponents, ...
0
votes
2answers
66 views
Exponentation vs Power
What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power?
Is any difference between them?
3
votes
3answers
154 views
Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$
$$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$
I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
1
vote
0answers
201 views
upper bound of exponential function
I am looking for a tight upper bound of exponential function (or sum of exponential functions):
$e^x<f(x)$ when $x<0$
or
$\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)$ when $x_i<0$
Thanks a ...
8
votes
2answers
299 views
Is there another representation for $x^x$
I started wondering about this the other day. Since the following have their own alternate representations. $$\begin{align*} \displaystyle\large x+x=2x & \
\frac{x}{x}=1 & xx=x^2\end{align*}$$
...
1
vote
1answer
46 views
Conventions for notation of function exponentation.
I read a previous question here but it seems incomplete for me (missing references).
Given a generic function, $ f $ :
1. is true that $ f^2 $ means $ f^2(x) = (f \circ f)(x) = f(f(x)) $ ?
2. or is ...
1
vote
0answers
94 views
Solving for $x$ in $y=x^x(\ln x + 1)$ (Lambert W?)
I made a bunch of problems exercising the Lambert W-function in the solution, because I like to exercise to new concepts that I learn about. One that I came up with was rearranging $y = x^x(\ln x + ...
2
votes
0answers
170 views
exp(ab) decomposition
How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or ...
2
votes
2answers
48 views
regarding exponents, how to interpret and use.
Some times in the books both of the below mentioned concepts are used interchangeably. Is there any reason for that? When to use -(2)^2 = -4 and (-2)^2. Explain with any useful examples.
30
votes
11answers
2k views
How is $e^x$ read aloud?
My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x.
How do you read aloud $e ^ x$?
Is it:
e raised to x
e power x
e powered x
or e ...
1
vote
2answers
83 views
Failure during calculating the matrix exponential, but where?
I have to calculate $e^{At}$ of the matrix $A$. We are learned to first compute $A^k$, by just computing $A$ for a few values of $k$, $k=\{0\ldots 4\}$, and then find a repetition. $A$ is defined as ...
1
vote
0answers
34 views
Triangular exponentation logarithm and inverse
The generalized formula of triangular exponentation on real numbers field is
$x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $
It's my ...
6
votes
7answers
189 views
Solving $1.1^n = n^{100}$
How do I go about solving for $n$ in the following equation:
$$(1.1)^n = n^{100}$$
A hint suffices.
0
votes
1answer
59 views
Reducing the Index and Improper Fractions
I'm trying to do the problem
3√40x^4/y^9.
When you try to reduce the index for 40^4, its going to be 4/3. How does the index get reduced into 2x√5x? I understand 3 cubed of 40, but what happens to ...
4
votes
2answers
81 views
Approximation with 1-exponential
How come that
$$\left(1-\frac{1}{x}\right)^x \approx e^{-1}\ ?$$
Is there a proof or something to understand this?
-4
votes
1answer
82 views
Boundedness on strips in the complex plane for functional equations [closed]
We know that the recurrence for $b>0$
(1) $f(0)=1$
(2) $f(z+1)=b{f(z)}$
has $f(z)=b^z$ as the only entire solution that is bounded on the strip $S=\{z: 0<\Re(z)\le 1\}$.
The image of $S$ ...
0
votes
2answers
54 views
solving in x involving both exponential and logarithmic function
Is it possible to solve a function with both exponential and logarithm such as
$a x^2−b.\log(x)= c$
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
5
votes
2answers
300 views
A “Matrix Trigonometry”
$e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix):
$$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$
A thought occurred to me that ...
1
vote
0answers
41 views
Is an equation of the following form solvable?
Is it possible to solve for $x$ which satisfies the equation $$d=(a\exp(bx)+\exp(cx))x^2$$ where $a,b,c,d$ are given constants? It looks quite horrible... Many thanks!
21
votes
1answer
275 views
Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
0
votes
3answers
100 views
Manipulating Exponents
I'm doing my homework and there are a couple of things that I am having trouble grasping. All my homework asks is that I simplify the exponents. For example: ...
6
votes
4answers
237 views
What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?
I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
2
votes
1answer
112 views
Cardinal exponentiation problem from Halmos' Naive Set Theory
In chapter 24 of Halmos' Naive Set Theory the following problem is posed as an exercise (page 96):
Prove that if $a, b$ and $c$ are cardinal numbers such that ${a}\le{b}$, then $a^c\le{b^c}$. ...
