Questions tagged [exponential-function]

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

The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance).

Definition: If $~b~$ be any number such that $~b\gt 0~$ and $~b\neq 1~$ then an exponential function is a function in the form,$$f(x)=a~b^x$$ where $~b~$ is called the base , the exponent,$~x~$ can be any real number and $~a\neq0~$.

${}$

Properties:

  • The graph of $~f(x)~$ will always contain the point $~(0,1)~$. Or put another way, $~f(0)=1~$ regardless of the value of $~b~$.
  • For every possible $~b~$we have $~b^x\gt 0~$. Note that this implies that $~b^x\neq 0~$.
  • If $~0\lt b\lt 1~$then the graph of $~b^x~$ will decrease as we move from left to right. Check out the graph of $~\left(\frac{1}{2}\right)^x~$ above for verification of this property.
  • If $~b\gt 1~$ then the graph of $~b^x~$ will increase as we move from left to right. Check out the graph of $~2^x~$ above for verification of this property.
  • If $~b^x=b^y~$, then $~x=y~$.

${}$ The Natural Exponential Function: In mathematics, the natural exponential function is $$f(x)=e^x~,$$ where $e$ is Euler's number.

Note: $f(x)=e^x~$ is a special exponential function. In fact this is so special that for many people this is THE exponential function.

Applications:

Exponential functions are solutions to the simplest types of dynamical systems. It is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative.

References:

https://en.wikipedia.org/wiki/Exponential_function

http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U18_L1_T1_text_final.html

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Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
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Evaluating $|a^b|$ when $a,b$ are complex

Here, $a^b=e^{b\log a}$ for some suitable (but fixed in advance) branch of the $\log$ function. What is the most general formula for $|a^b|$ when both $a$ and $b$ are complex, and what are the conditions on it? The formula $|a^b|=a^{\Re b}$ only…
Mario Carneiro
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Simplifying expression and finding indefinite integral

(a) Simplify $$\Large \frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \quad.$$ (b) Hence find $$\Large \int\frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \mathrm{d}x$$ I tried to find a breakdown of the expression, but I have no clue how to simplify that. I can do the…
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Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. Note: I apologize for the terrible formatting, I just…
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Summation of exponential series

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
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Why does a heating model work?

I am referring to: $T=T_0 e^{kt}$ where T=temperature,t=time and k=constant. It seems to work, I as just curios to why it works?
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If $\ln x$ is defined via an integral and $e$ defined from $\ln x$, how would you prove that $\ln x$ is the inverse of $e^x$?

This is a somewhat technically specific question about the relationship between $\ln x$ and $e^x$ given one possible definition of $\ln x$. Suppose that you define $\ln x$ as $$\ln x = \int_1^x{\frac{dt}{t}}$$ We can use the connection between the…
templatetypedef
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Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
StudentsTea
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Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma function Can anyone help me how to solve it? Thank…
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Evaluating exponential integral

I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ $N$ is a positive integer, $n$ is an integer, $\alpha$ can be a…
divB
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Why is sequence $(1+\frac{1}{n})^{n+1}$ descending?

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why is that? On what $n$ this happens? Edit: Let me…
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Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density function of the standard normal distribution. I…
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How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r \log{a}}$ but it doesn't help me. Thanks
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Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not true. Writting $a^x=\exp(x\log(a))$ $(E)$ can be…
user142836
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Integral of a sum of complex exponentials

Let $$\hat{\varphi_n}(t)=\frac{1}{n}\sum_{j=1}^n{exp(i{t}Y_j)}\quad(t\in\mathbb{R})$$ denote the empirical characteristic function of the residuals $Y_j\,=\,S_n^{-\frac{1}{2}}(X_j-\bar{X}_n),\quad j=1,\dots,n,$ where…
Bo25
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