1
vote
3answers
28 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
-1
votes
3answers
38 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
0
votes
1answer
20 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
-8
votes
2answers
82 views

I need to solve this equation [do my homework]. [closed]

Problem for my university math class 1st semester Thanks a lot!
0
votes
1answer
15 views

exponential equation (solve for x)

We started review in my calculus class and I have mostly forgotten everything about exponential equations. $$ e^{-x} * (3e^{2x}-(5/4))^{1/2} = e^x $$ Would x just equal 0, when -x+x=0 (two sides of ...
6
votes
1answer
75 views

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
0
votes
1answer
42 views

Big-Oh of exponent of exponent

How does one whether an exponent of an exponent is the big-Oh of the other? For example, if I have $a^{b^n}$ and $b^{a^n}$, how would i determine and prove which is a big oh of another? I'm thinking ...
0
votes
2answers
60 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
3
votes
3answers
71 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
0
votes
1answer
41 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
1
vote
1answer
48 views

$\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
0
votes
1answer
45 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. ...
0
votes
1answer
22 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 ...
0
votes
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 ...
0
votes
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
votes
4answers
96 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 ...
1
vote
1answer
49 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 ...
1
vote
2answers
39 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
-1
votes
1answer
69 views

how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
3
votes
1answer
71 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
0
votes
2answers
32 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
0
votes
0answers
16 views

How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
2
votes
1answer
56 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
2
votes
7answers
130 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
0
votes
1answer
44 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
18
votes
4answers
312 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
3
votes
2answers
105 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
3
votes
4answers
153 views

Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
3
votes
1answer
61 views

A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
10
votes
1answer
188 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
0
votes
0answers
50 views

Are the solutions to this integral known?

Mathematica knows that the real part of $y$ in this integral: $$\int_0^{\infty } \frac{1}{x^{1/y}+2} \, dx$$ is: $$0<\Re(y)<1$$ Therefore I am wondering if the solutions to this integral known? ...
2
votes
0answers
45 views

Exponential equations solving methods?

Do you have an idea or general method to solve the following equation?: $$a^{\alpha x}+b^{\beta x} = c^{\gamma x}+ d^{\delta x}$$ when $a,b,c,d$ aren't zero, and $\alpha, \beta, \gamma, \delta$ are ...
0
votes
1answer
90 views

Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
0
votes
3answers
80 views

Complex exponential

I know that the equation $e^{z}=-1$ has no solution had if been $z$ is a real number. So does the equation also has no solution when $z$ is complex?
2
votes
2answers
84 views

Differentiation of exponential function? [closed]

How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $$ {}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ...
1
vote
1answer
45 views

Solving for exponent with multiple bases

From a practical perspective, my question can be most easily considered as solving for time in a future-value type equation, but for two separate investments growing at different rates. Say you have ...
0
votes
1answer
63 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
2
votes
1answer
99 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
5
votes
2answers
40 views

Upper bound for product of exponents

From here we have the bound $$\left(1-\frac1N\right)^N\leq e^{-1}$$ where $N$ is a positive integer. Written another way, it is ...
0
votes
0answers
117 views

Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$ Even though there are a lot of integrals involving the ...
0
votes
0answers
96 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
0
votes
1answer
42 views

Calculate the Burning Time for a Lamp

If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ...
1
vote
0answers
863 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
0
votes
1answer
59 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
0
votes
1answer
32 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
4
votes
1answer
134 views

Question involving exponential tower of 19

Consider: $$ y = \underbrace{19^{19^{\cdot^{\cdot^{\cdot^{19}}}}}}_{101 \text{ times}} $$ with the tower containing a hundred $ 19$s. Take the sum of the digits of the resulting number. Again, add the ...
5
votes
3answers
421 views

Solve an equation with $e^{(x-2)}=e^{4}\cdot e^{\sqrt{x}}$

$$e^{(x-2)}=e^{4}e^\sqrt{x}$$ I know that $x = 9$ and I can show the calculations like this: $$e^{(x-2)} = e^{\sqrt{x}+4}$$ and now I need to get the $x$ to the right side but I dont know how.
0
votes
0answers
96 views

An equation to represent an exponential increase from 0 to 1, weighted towards 1

I calculated an equation to represent agreement between different sources in an historical study: $$ 1 - \frac{1}{ \left(\frac{y^2+1}{x^2+1}\right) } $$ It gives me a nice table like this: ...
5
votes
1answer
65 views

Do we really know the value of expressions with irrational powers?

The way we evaluate decimal powers such as $a^.75$ is by splitting it into $(a^3)$^(1/4). How then can we evaluate irrational powers? I know that we can approximate, but whenever we graph a^x we ...