For question involving exponential functions and questions on exponential growth or decay.

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4answers
25 views

Half-life of Am-$241$, $3$ micrograms decays over $9$ years, how much if left?

$3$ micrograms of Americium-$241$, which has a half life of $432$ years. After $9$ years how much will remain? I'm not sure of the formula to use or how to calculate it. I'm assuming it's exponential ...
0
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1answer
28 views

How to solve $ax+be^{cx}=d$

Hi I have been trying to derive the formula for the range of a projectile with air resistance, and I've been trying to solve for time when $y=0$, and I have been left with an equation of the form ...
0
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1answer
32 views

Newton's Law of Cooling (and Heating)

The Formula for the equation is as follows: $$T(t)=\frac {\int^t(−T_s)ke^{-kt'}dt'+C}{e^{-kt}}$$ This formula is needed to determine the temperature at time $t$, $T(t)$, of an object as it begins to ...
2
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2answers
52 views

Prove that $e$ is the root of the equation $\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$

It seems numerically that $e$ is the only real root of the equation: $$\int_0^{\infty} \frac{dt}{(t+x)\sqrt{4t+(x+1)^2}}=\frac{1}{x-1}$$ Mathematica confirms it at least to the large number of ...
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1answer
25 views

Newtons Law of Cooling in Forensic Science

Question goes: Law enforcement would like to know the time at which a person died. The investigator arrived on the scene at 8:15pm, which we will call $t$ hours after death. At 8:15 (i.e $t$ hours ...
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2answers
32 views

Questions about Exponentiation and roots and logarithms.

in this page a few questions I want to ask you about the Exponentiation and roots and logarithms: What and how the Exponentiation definition can be defined by real numbers.? What is the overall ...
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4answers
254 views

Finding limit without using limit

If we have to find the value of $$ \lim_{x \to 0} \frac{e^x-1}{x}$$ I tried to solve this by using series i.e by expanding $e^x$ and got the result. But if there is another method to solve this
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0answers
22 views

Calculation of an integral involving the sum of a range of natural exponential functions

Does somebody know how to solve the following integral, I extremely hope I can obtain its close-form solution: \begin{equation} \int \sqrt{ \sum_{i=1}^{M}\sum_{j=1}^{M} ...
2
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0answers
19 views

Integral of third order polynomial exponential

I am looking for approximated or exact solution of \begin{align} I = \int_R \exp(cx^3-ax^2+bx)dx \end{align} where $a,b,c$ are complex numbers defined as: \begin{align} c &= ...
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2answers
35 views

calculate a natural exponential equation

Hi I try to solve the following equation $50e^{0.15t} - e^{0.98t} = 38$ where $0.15t$ and $0.98t$ are the indices of e Can someone tell me how to derive the value of $t$ step by step? I am also ...
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2answers
73 views

How do you solve x^2 = log^2(x)

I read a page that said that the limit as $x$ approaches infinity of (polynomial function)/(logarithmic function) = infinity and that the limit as $x$ approaches infinity of (logarithmic ...
2
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3answers
65 views

Exponential Fundamental Limits without using L'Hôspital's rule

I have a limit to evaluate. $$\lim_{x\to2} \left(\frac{\mathrm e^x - \mathrm e^2}{x-2}\right)$$ Can someone solve it without using L'Hôspital and explain me the steps? Thanks
4
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6answers
88 views

Find $ \lim_{x\to 0} \left(\frac {\tan x }{x} \right)^{\frac{1}{x^2}}$.

Can someone help me with this limit? I'm working on it for hours and cant figure it out. $$ \lim_{x\to 0} \left(\frac {\tan x }{x} \right)^{\frac{1}{x^2}}$$ I started transforming to the form $ ...
6
votes
1answer
93 views

Is there any nice explanation of why the complex exponential function has no roots in the complex plane? [duplicate]

Here I am not looking for an explanation that uses basic properties that complex exponential function has, such as $e^{z+w}=e^ze^w$ or $e^0=1$ or any other, if this fact can be explained by using ...
0
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1answer
29 views

Solving for a negative exponent

Is it possible to solve for a negative exponent? If so, can someone help me get the n on one side of the equation? I'm not a math student and I have no math teacher connections so I thought I would ...
0
votes
1answer
29 views

How long will it take the number of bacteria to double?

The number of bacteria in a strain is given by $B(t) = 30e^{1.5t}$, where $t$ is the time in hours. a) How many bacteria are there at time zero? b) How long will it take the number of ...
0
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1answer
28 views

Find the half life using exponential expression

a) For a particular radioactive substance, the mass $m$ (in grams) at a time $t$ in years is given by $m = m_0e^{-0.02t}$, where $m_0$ is the original mass. If the original mass is $500$g, ...
14
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5answers
224 views

How to show $\frac{19}{7}<e$

How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...
0
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1answer
32 views

Rewriting $n(2^{2^{n-1}}-1)-2^{2^n}+2^{2^{n-1}+1}$

Could you help me to write in a better way the following expression? (by better I mean for example simplifying if I can) $$ n(2^{2^{n-1}}-2)-2^{2^n}+2^{2^{n-1}+1} $$
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2answers
27 views

Find base of exp function within the range of summation

I got a sequence where the relation between elements of the sequence is given by: \begin{align} y_1 &= b \\ y_{i+1} &= 2 y_i + b \quad (i \in \mathbb{N}) \end{align} where $b$ is called base, ...
0
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0answers
16 views

Make a formula based on a data table (Exponential function)

I always, since high school never found a good trick to do these kind of questions. Lets say you've got a table (x and y) X: 1 2 3 4 5 Y: 1 3 7 15 31 How can I make a function out of it? I ...
2
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0answers
39 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
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3answers
86 views

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$?

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$? So far I have seen the following results: If $f\colon U\to\mathbb C$ has an antiderivative $F$ on $U$ then ...
1
vote
1answer
19 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
0
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1answer
33 views

Solving equation in Matlab [closed]

I want to solve this equation for several values of y y=4 exp(0,034x) y= [60 14 63 34 21 12 11]; Thanks
1
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1answer
30 views

Evaluating a tricky exponential function integral

I am trying to evaluate the following integral $$ I = \int_0^t s^{2\alpha - 1} \exp\left(\frac{i \sqrt{2} \left(t^{2 \alpha + 1} - s^{2 \alpha + 1}\right)}{ 2 \alpha + 1}\right)\mbox{d}{s} $$ where ...
2
votes
1answer
30 views

Help in evaluating an integral of exponential function

I am trying to evaluate the following integral $$ I = \int_{0}^{t}s^{-b-1}e^{-\frac{1}{2} a^2 s^{-2 b}} ds$$ where $a > 0$ and $ 0 \le b \le 1$. I am not quite sure how to solve this. Any help ...
2
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3answers
36 views

Minor flaw in understanding of the proof of the derivative of exponential functions

I understand the majority of the proof of the derivative formula for exponential functions of the form: (full proof at bottom of post) $\frac{d}{dx}a^x$ but I have a little trouble with the last ...
2
votes
5answers
188 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
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1answer
33 views

Functions of the form $f(x) = k^x - x^k$

Let $f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = k^x - x^k$ where $k \in \mathbb{R}$ is a given constant. Currently I am thinking of positive $k$ and positive $x$ because there would be complex ...
5
votes
3answers
494 views

How much proof is needed in such paper (Maths related)?

I'm writing a paper (report) regarding Euler's Number $\space e \space$ (even though he didn't discover it). Within this paper, I show that: $${d\over dx} {e^x} = {e^x}$$ **NOTE: ** This is not ...
0
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1answer
18 views

The upper bound of a sum of exponential function

Could someone help me to find the upper bound of the following function: $f(x) = \sqrt{\sum_{n=i}^{N} e^{-\alpha_{i}\cdot x}}$, where $x > 0$, the $i^{th}$ coefficient $\alpha_{i} > 0$. I got ...
1
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0answers
22 views

Formal Power Series Composition with Exponential

I have seen formal power series expressed as $$B(z) = \sum_{i=1}^{\infty} b_i{z}^i,$$ but then also as $$B(z) = \sum_{i=1}^{\infty} \frac{b_i}{i!}{z}^i.$$ Is there a significant difference between ...
0
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2answers
70 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...
1
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1answer
23 views

Exponential Decay of a radioactive substance

If $375$ mg of a radioactive substance decays to $300$ mg in $72$ hours, find the half-life of the element. I first used the mathematical formula of $$A = A_0e^{kt}$$ or exponential decay. After ...
1
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0answers
44 views

A variant of the exponential integral

Consider the following integral (for $x,y\in \mathbb{R}_{>0}$) $$E(x,y) = \int_0^1 \frac{\mathrm{e}^{-x/s-ys}}{s}\,\mathrm{d}s,$$ which is a variant of the usual exponential integral $E_1(x)$ to ...
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0answers
8 views

Name for this type of function form.

I use the function form of $$f(x)=ln^{2}(1+e^{(Ax)})$$ to denote the different between operating regimes of the MOSFET. This form let's me glue the diffusion and drift current of the MOSFET I-V ...
0
votes
1answer
14 views

Finding the modulus of complex functions

Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$ I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$ To find ...
0
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0answers
62 views

Very difficult functions to prove with O notation

I am trying to prove some O notations as is it one of the tasks for my assignment in my course in algorithms and data structures. First of all I'd like to be sure that I got the "recipe" right. I use ...
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1answer
39 views

Proving that: $e^{-x(1/\tau - i\xi)} \to 0$ as $x \to \infty$.

I remember my friend showing me how sandwich theorem can be applied here. Unfortunately, I can't find his solution anymore and I am not familiar with sandwich theory.
1
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0answers
36 views

Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
1
vote
2answers
74 views

Question about the proof of Central Limit Theorem

My instructor proved the central limit theorem using the characteristic function. I think the proof is a standard one because I found basically the same proof in wikipedia. So for i.i.d. ${X_1, ...
0
votes
1answer
47 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
0
votes
1answer
21 views

finding the growth rate for exponential growth

I have this question, Determine the initial population of a bacterial culture whose growth is exponential if, after $7$ days, the population is $10$ million, and the number triples every in three ...
0
votes
1answer
44 views

Complex exponential with 2 pi

I wonder why is it wrong to do the following: $e^{i2\pi x}=(e^{i2\pi})^x=1^x=1$ for a real $x$ but not for an integer $x$
1
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1answer
21 views

Exponential form of a log

I'm a bit confused on the wording of this question: An equation is shown below x = log(20) What is the exponential form of this equation? So my answer is $10^x$=20. But I am not sure if that is ...
1
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1answer
26 views

Is the set $\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$open in $TM$?(where $r_p$ the injectivity radius at $p$)

Let $(M,g)$ is a Riemannian manifold. (1)If $D_p$ is the largest domain on which $\exp_p$ can be a diffeomorphism, then is the set $$D=\bigsqcup_{p\in M} D_p$$ open in $TM$? (2)Likewise, if we denote ...
0
votes
1answer
18 views

Explanation of homogenous function

Is there someone, who can explain why the function $g(s)=f(e^s,e^s)$ is not homogeneous when it can be written as $\frac{9}{4}e^{s/2}s$. I got the function $f(x,y)=\sqrt x +2\sqrt y +\frac{3y}{\sqrt ...
15
votes
3answers
2k views

Why is Euler's number used as a base for logarithms? [duplicate]

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms? Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't ...
-1
votes
3answers
69 views

Find a solution to $z+e^{-z}=a$ where $a>1$.

Find a solution to $z+e^{-z}=a$ where $a>1$. I have tried many manipulations with little success. I don't see how I can solve this for $z$. Any solutions or hints are greatly appreciated. I think ...