# Tagged Questions

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### 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 ...
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### $\mathrm{Ei}(x)$, 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 ...
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### 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. ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $|x|^{|x|}$ 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n\times n$ ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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?
How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $${}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ... 1answer 45 views ### Solving for exponent with multiple bases From a practical perspective, my question can be most easily considered as solving for time in a future-value type equation, but for two separate investments growing at different rates. Say you have ... 1answer 58 views ### Solve some unusual log/exponential equations I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course):  4^x = x + 10 x^x = 3 (2x + 3x^2)^{x + 1} = (x - x^3)^{x^2} Are ... 1answer 78 views ### Trouble solving polynomial equation with exponent I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me?$$9x^4-13x^2+4 = 0$$2answers 37 views ### Upper bound for product of exponents From here we have the bound$$\left(1-\frac1N\right)^N\leq e^{-1}$$where N is a positive integer. Written another way, it is ... 0answers 109 views ### Integral involving Modified Bessel function, exponential and power I am trying to evaluate the following integral:$$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$Even though there are a lot of integrals involving the ... 0answers 89 views ### Development of imaginary exponent without appealing to “ambiguity” between i and -i Is there a way to develop the definition of the imaginary exponent, z^i, for complex z, that does not appeal to the notion that i and -i are "qualitatively indistinct" and that does not rely ... 1answer 41 views ### Calculate the Burning Time for a Lamp If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ... 0answers 695 views ### How to solve polynomial-exponential equation I'm trying to solve equations like the following one:$$5 + 3x - 4x^3 = e^{x^2}$$I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ... 1answer 58 views ### Relation of e to other numbers… I found the following result, When i was working on my calculator .$$x^y < y^x \quad ,x < y \quad \text{ for } x,y<ex^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$I can't ... 1answer 32 views ### Growth rate of n^2 vs (\log_3(n))^3 Which grows faster, n^2 or (\log_3(n))^3? How do I figure out which grows faster in general in these kinds of situations? 1answer 124 views ### Question involving exponential tower of 19 Consider:$$ y = \underbrace{19^{19^{\cdot^{\cdot^{\cdot^{19}}}}}}_{101 \text{ times}} $$with the tower containing a hundred  19s. Take the sum of the digits of the resulting number. Again, add the ... 3answers 405 views ### Solve an equation with e^{(x-2)}=e^{4}\cdot e^{\sqrt{x}}$$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}$$and now I need to get the x to the right side but I dont know how. 0answers 91 views ### An equation to represent an exponential increase from 0 to 1, weighted towards 1 I calculated an equation to represent agreement between different sources in an historical study:$$ 1 - \frac{1}{ \left(\frac{y^2+1}{x^2+1}\right) } $$It gives me a nice table like this: ... 1answer 63 views ### Do we really know the value of expressions with irrational powers? The way we evaluate decimal powers such as a^.75 is by splitting it into (a^3)^(1/4). How then can we evaluate irrational powers? I know that we can approximate, but whenever we graph a^x we ... 1answer 675 views ### Can you raise \pi to a real power to make it rational? We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this: Take (\sqrt{2})^{\sqrt{2}} If it's rational, ... 0answers 85 views ### Generalisation of Lambert W function? I want to solve an equation of the form: \exp(C / x) - 1 = D / (x + a) This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ... 6answers 116 views ### Motivation for creation of complex exponentiation I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived? 2answers 145 views ### upper bound for e^{ax^2} I want to find a upper bound for$$e^{ax^2}\leqslant \: ?$$"a" is a constant and a\geqslant 0 . x is a variable. I prefer to have a polynomial function or power function (like  x^{k}) is there ... 1answer 81 views ### Why do we use the form f(t)=ae^{kt} for exponential growth and decay? Why do we include the e^{k}? Wouldn't it be easier to simply use f(t)=ap^{t} where p is the percentage increase per time. Is there a reason why the convention is to use f(t)=ae^{kt}? 18answers 2k views ### How to understand why x^0 = 1, where x is any real number? Alright, so the idea of an exponent, x, is that you are multiplying its base by itself x number of times. With base 5 and x=3, we have that 5^3 = 5 \cdot 5 \cdot 5 I understand that the ... 1answer 103 views ### How do I solve this exponential equation?$$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}$$3answers 182 views ### Solve 3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$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?
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 ...