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

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2
votes
1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
0
votes
0answers
16 views

Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$? for any $j\in \mathbb{N}$, where $\delta<1$

For $N\in\mathbb{N}$ large enough. Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$ for any $j\in \mathbb{N}$, where $\delta<1/2$? I tried using Matlab for j=5, but I'd like some ideas for a proof. ...
0
votes
0answers
21 views

Is $(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$ really true?

In the paper "Preserving Statistical Validity in Adaptive Data Analysis", it says that if $p \in (0,1], \tau \in [0, 1/3]$ then $$(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$$ I understand that $(1 + \tau ...
0
votes
2answers
31 views

Calculating the mass xkg of radio-active substance pertaining to days after starting timing

Just testing myself with some tricky questions in my further maths textbook. This one states that the mass xkg of a radio-active substance remaining in a sample t days after starting timing is given ...
0
votes
2answers
28 views

multiplying powers with variable in exponent and different bases

I am having trouble sorting out where to begin with solving for unknown value in this equation: $16^{5a−1} \times 256^{3a} = 128$. I imagine I would need to change to logarithmic form, but am ...
0
votes
1answer
59 views

If $x>0$ and $x^a=x^b$ can one assume that $a=b$?

If $x>0$ and $x^a=x^b$ can one assume that $a=b$? The answer says it's not right. I've tried coming up with a counter-example but keep failing. Thanks in advance!
1
vote
2answers
39 views

$(5 + (24)^{\frac{1}{2}})^x + (5 - (24)^{\frac{1}{2}})^x = 10$ , solve for $x$

I have been stuck to this question lately $(5 + \sqrt{24})^x + (5 - \sqrt{24})^x = 10$ , solve for $x$
-1
votes
1answer
19 views

fill in a table with a 7% yearly growth [closed]

There is $1000$ put in a account and it grows $7\%$ each year starting at $1000$. I'm confused on how to get the calculation and I have several problems just like this.
0
votes
0answers
25 views

Coefficient Correlation r of Exponential Functions Regression

I'm writing an exponent regression calculator $Ae^{Bx}$ Sample Data Set (X,Y) is (9, 1) (7, 10) (6,11) (20, 10) (15, 1) A = 5.287 and B = -0.0232. So $F(x) = 5....
4
votes
2answers
66 views

Prove $\exp(\mathrm{Tr}(X))=\det(\exp(X))$

Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) \times\...
0
votes
0answers
30 views

Shift logistic function without moving inflection point from x=0

As a biologist that did not do much maths lately, formulation of my problem may be a bit strange. Sorry in advance and thanks for helping me improving my vocabulary. I am using logistic function from ...
0
votes
1answer
52 views

How do you solve a system of equations with e^x

How do you solve a system of equations with e^x. For example
3
votes
0answers
108 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
1
vote
2answers
42 views

Combining linear and exponential functions

I am struggling with this problem: At $x=0$ I own $b$ units. Every year I deposit $a$ units. The bank pays an interest rate of $c$ every year. After how many years I will own $d$ units? $$(ax+b)*e^{...
0
votes
4answers
63 views

Exponential decay 'proof' that $.\overline{9}\neq 1$?

I have doubts about $.\overline{9}$ being equal to 1 due to the following proof: To get a decimal containing $c$ 9's after the decimal point, the equation f(c) = $1-10^{-c}$ can be used. For ...
0
votes
3answers
47 views

Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
-1
votes
2answers
32 views

Simplifying a term

I want to know how we got the end term. It may be trivial, but I just can´t figure it out. Thanks for the help. Term
1
vote
1answer
45 views

Symmetric Matrix in SO(3) : Exponential Formula

Let $R\in $ SO(3), that is $R$ is real $3\times 3$ orthogonal matrix with determinant $+1$. I am trying prove that if $R= R^\top$, and $R\in $ SO(3) then $R \in \{exp(k\pi \hat{a}) | k\in \mathbb{Z}, ...
0
votes
1answer
42 views

How to compute the exponential of this matrix?

I am trying to prove all the results regarding linear algebra in my ODE class. I have already convinced myself that if I have a matrix $T$ which has an eigenvalue $\lambda = a + ib$ and an associated ...
1
vote
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
votes
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 $at+...
0
votes
1answer
35 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
votes
2answers
56 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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
4answers
255 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
0
votes
0answers
23 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} e^{-\frac{\frac{\left|\mathbf{...
2
votes
0answers
21 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 &= \frac{1}{3}i\pi\phi'''...
1
vote
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 ...
0
votes
2answers
74 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 function)/...
2
votes
3answers
67 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
votes
6answers
89 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
99 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
votes
1answer
30 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
32 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
votes
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, find ...
14
votes
5answers
238 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
votes
1answer
33 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) $$ n(2^{2^{n-1}}-2)-2^{2^n}+2^{2^{n-1}+1} $$
0
votes
2answers
28 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
votes
0answers
18 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
votes
0answers
41 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! \...
0
votes
3answers
88 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 $\displaystyle\int_\...
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
votes
1answer
34 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
vote
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
votes
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
227 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
vote
1answer
37 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
496 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 ...