What is Exponential: Definition and 1000 Discussions
In mathematics, the exponential function is the function
f
(
x
)
=
e
x
,
{\displaystyle f(x)=e^{x},}
where e = 2.71828... is Euler's constant.
More generally, an exponential function is a function of the form
f
(
x
)
=
a
b
x
,
{\displaystyle f(x)=ab^{x},}
where b is a positive real number, and the argument x occurs as an exponent. For real numbers c and d, a function of the form
f
(
x
)
=
a
b
c
x
+
d
{\displaystyle f(x)=ab^{cx+d}}
is also an exponential function, since it can be rewritten as
a
b
c
x
+
d
=
(
a
b
d
)
(
b
c
)
x
.
{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}
The exponential function
f
(
x
)
=
e
x
{\displaystyle f(x)=e^{x}}
is sometimes called the natural exponential function for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since
a
b
x
=
a
e
x
ln
b
{\displaystyle ab^{x}=ae^{x\ln b}}
As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
d
d
x
b
x
=
b
x
log
e
b
.
{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}
For b > 1, the function
b
x
{\displaystyle b^{x}}
is increasing (as depicted for b = e and b = 2), because
log
e
b
>
0
{\displaystyle \log _{e}b>0}
makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1/2); and for b = 1 the function is constant.
The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:
This function, also denoted as exp x, is called the "natural exponential function", or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as
b
x
=
e
x
log
e
b
{\displaystyle b^{x}=e^{x\log _{e}b}}
, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression. The graph of
y
=
e
x
{\displaystyle y=e^{x}}
is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation
d
d
x
e
x
=
e
x
{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}
means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point. Its inverse function is the natural logarithm, denoted
log
,
{\displaystyle \log ,}
ln
,
{\displaystyle \ln ,}
or
log
e
;
{\displaystyle \log _{e};}
because of this, some old texts refer to the exponential function as the antilogarithm.
The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well):
It can be shown that every continuous, nonzero solution of the functional equation
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
{\displaystyle f(x+y)=f(x)f(y)}
is an exponential function,
f
:
R
→
R
,
x
↦
b
x
,
{\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},}
with
b
≠
0.
{\displaystyle b\neq 0.}
The multiplicative identity, along with the definition
e
=
e
1
{\displaystyle e=e^{1}}
, shows that
e
n
=
e
×
⋯
×
e
⏟
n
factors
{\displaystyle e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ factors}}}}
for positive integers n, relating the exponential function to the elementary notion of exponentiation.
The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (e.g., matrix).
The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions 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. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
Let K be any Matrix, not necessarily the hamitonian. Is $$e^{-Kt}\left|\psi\right>$$ equal to $$e^{-K\left|\psi\right>t}$$ even if it is not the the eigenvector of K?
I think so as i just taylor expand the $$e^{-Kt}$$ out but I want to confirm.
In that case can i say that...
Homework Statement
a= eD/R*T*G make a linear equations
and calculate the value for D and G
R=8,3 and constant
D,G=constant
T= variable
Homework Equations
y=ax+b
y=numberax*bThe Attempt at a Solution
ax= E/(R*T)
x= 1/T
a= E/R
y= (E/R)*x+G
I don't know how to move on and if this is even correct/
Homework Statement
I think I am being stupid, I am trying to show that
## \int^{T}_{0} e^i\frac{2\pi(n-m)t}{T} dt = 0 ## [1] if ## n \neq m##
## = T ## if ##n=m##, ##T## the period.
Homework Equations
[/B]
I am using the following ##cos## and ##sin## orthogonal...
Homework Statement
[/B]
Solve the equation
## e^{2x}+2=e^{3x-4}##Homework EquationsThe Attempt at a Solution
I know by using Newton-Raphson method the problem can be solved, i however tried solving it as follows
##e^{3x-4}-e^{2x}-2=0, e^{3x}-e^{2x}.e^{4}-2e^{4}=0, p^3-p^2.e^4-2e^4=0...
I have a simple complex exponential signal of the form x(t)=ejωt. To find period of the signal I tested if x(t)=x(t+nT) for all n:
ejωt=ejω(t+nT) ⇒ ejωnT=1=ej2πk
where n and k are integers. Then I find a general period expression as
T=2πk/ωn
Period T means it is the least time a signal...
1. Homework Statement
Find derivative of
y=e^(cos(t)+lnt)
Homework EquationsThe Attempt at a Solution
So just using the chain rule:
y'=e^(cos(t)+lnt)*(-sin(t)+1/t)
The answer in the back of the book is
y'=e^(cos(t))*(1-tsin(t))
I want to create a binary matrix (${X}_{m*n}$) containing $C$ ones ($||X|| = C$). Additionally I want to have the number of elements of each row ($m$) to have an exponential form. This is: for each row the number elements needs to be equal to $e^{alpha * i}$ or in symbols: ||${X}_{i}$|| =...
Homework Statement
If ##r_1, r_2, r_3## are distinct real numbers, show that ##e^{r_1t}, e^{r_2t}, e^{r_3t}## are linearly independent.
Homework EquationsThe Attempt at a Solution
By book starts off by assuming that the functions are linearly dependent, towards contradiction. So ##c_1e^{r_1t}...
Homework Statement
I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations
Only that ## e^{x} =...
The wave function is an exponential function, if I plot the real part of it, I don't get a wave graph like sine or cosine function, Why the wave function is not represented by a trigonometric ratio instead.
Also, the wave function cannot be plotted since it is imaginary, why is it imaginary?
Thanks
hi everyone initially I really want to put into words that there is absolutely no source related to following probability in poisson process and distribution $$P(S^1_A<S^1_B<S^1_C)$$ or $$P(S^n_A<S^m_B<S^k_C)$$ where $$S^1_A = \text{first arrival of A event}, S^1_B= \text{first arrival of B...
Hi, I've been reading Roger Penrose's Road to Reality during my free time, and I am trying to do all the proofs I possibly can, although I am quickly reaching my limit.
Would somebody help me prove this?
h(x)=0 if x ≤ 0
h(x) = e-1/x if x > 0
Thank you in advance :)
Consider the operator Ô, choose a convenient base and obtain the representation of
\ exp{(iÔ)}
Ô =
\bigl(\begin{smallmatrix}
1 & \sqrt{3} \\
\sqrt{3} & -1
\end{smallmatrix}\bigr)
Attempt at solution:
So, i read on Cohen-Tannjoudji's Q.M. book that if the matrix is diagonal you can just...
So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials.
Here's the problem
A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of...
this is not a homework question, I just want to make sense of the equation here.
Assuming matrix A is diagonal,
If A_hat=T'AT where T' is an inverse matrix of T.
e^(A_hat*t)=T'e^(At)T
which implies,
e^(T'AT*t)=T'e^(At)T
we know that e^(At) is a linear mapping, therefore if we convert f to...
Mod note: Changed title from "Differential Euler's Number"
1. Homework Statement
Find the derivative.
f(t)=etsin2t
The Attempt at a Solution
f'(t)=etsin2t(sin2t)(cos2t)(2)
However the book seems to say that there should be an extra "t" in the solution. Some help?
Homework Statement
Using experimental data, I am trying to solve the following equation for ##T_2##
$$M(t)=M_0 e^{-t/T_2}. \tag{1}$$
Here ##M_0## denotes the initial value. There were 8 data points collected for ##t## and ##M(t).## Here is the resulting graph:
The data and equation (1)...
Homework Statement
Write the given numbers in the polar form ##re^{i\theta}##.
## \frac {2i} {(3e^{4+i})} ##
Homework Equations
## z = re^(i\theta) ##
## \theta = Arg(z) ##
## r = |z| = \sqrt { x^2 + y^2 } ##
The Attempt at a Solution
I'm not really sure how to go about the exponential...
Homework Statement
2x=1/ square root of x.
Homework Equations
[/B]
None that I feel ought to be included.
The Attempt at a Solution
To answer the question I graphed 2^x and 1/sqrt x individually. They intersect at an x value of 0.5, so x= 0.5.
The problem is that while the answer was...
Homework Statement
How is ## e^log√(1-x^2)## equal to ##√(1-x^2)?##Homework EquationsThe Attempt at a Solution
taking ln on the function, ln√(1-x^2). lne⇒ ln√(1-x^2) ....
Homework Statement
In 2003 the city of spring field had a population of 250000 the population is expected to double by 2025, how many people in 2015?
Homework EquationsThe Attempt at a Solution
A=Pb^t
The initial is 250000 and b is 2 because it doubles however I am unsure of what the exponent...
The problem
Solve ## e^x-e^{-x} = 6 ## .
The attempt
$$ e^x-e^{-x} = 6 \\ e^x(1-e^{-1}) = 6 \\ e^x = \frac{6}{(1-e^{-1})} \\ x = \ln \left( \frac{6}{1-e^{-1}} \right) \\ $$
The answer in the book is ## \ln(3 + \sqrt{10})##
Could someone help me?
Hi there! First Post :D
In a recent CM module we've been looking at coupled oscillators and the role of time translational invariance in the description of such physical systems. I will present the statement that I am having trouble understanding and then continue to elaborate.
In stating that...
Homework Statement
Homework Equations
A = P(1+i)^n
The Attempt at a Solution
Here is my attempted solution, can someone please verify if my method is correct!
Thanks in advance![/B]
Homework Statement
Homework Equations
The Attempt at a Solution
here is my attempted answer, can someone please verify if my method is accurate and if my solution is correct? thanks for the help!
Hi there. I was trying to implement Numerov method for solving second order differential equations (you can see some information of Numerov algorithm at here: https://en.wikipedia.org/wiki/Numerov%27s_method ). At the beginning I've used a uniform grid mesh, all points equally spaced. I'm...
Hello,
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'''(t) \notag\\
a &= \dfrac{1}{2\sigma^2}-i\pi \phi''(t) = re^{i\varphi}~~\text{with}~~~ r =...
If I rearrange an equation invoving exponentials of operators and I take ex to the opposite side of the equation it becomes e-x. What happens if I try to take eA to the opposite side ? I know a exponential of operators can be expanded as a Taylor series which involves products of matrices but...
Homework Statement
If ##a, b \in \{1,2,3,4,5,6\}##, then number of ordered pairs of ##(a,b)## such that ##\lim_{x\to0}{\left(\dfrac{a^x + b^x}{2}\right)}^{\frac{2}{x}} = 6## is
Homework EquationsThe Attempt at a Solution
So, this is a typical exponential limit...
Hello,
I have the following code in Mathematica, and it gives the following error:
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 12 recursive bisections in x near {x} = {0.000156769}. NIntegrate obtained 0.21447008480474927` and 5.637666621985554`*^-13 for the...
Homework Statement
Hi, I have come across this equation in modelling exponential growth and decay. I am wondering if it is possible to solve it algebraically or not?
Homework Equations
8000-1.2031 * e^ (0.763x)=(0.5992×e^0.7895x)
The Attempt at a Solution
Brought all e^(ax) values to one...
After following the logical steps to derive something, I reached to the following integral:
\int_0^{\infty}\frac{1}{x^2}\exp\left(\frac{A}{x}-x\right)E_1\left(B+\frac{A}{x}\right)\,dx
where ##E_1(.)## is the exponential integral function, and ##A## and ##B## are non-zero positive constants. I...
Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem:
∫ 1/(x^2+1)dx
I immediately...
In my electrical engineering textbook, they have an entire chapter devoted to the complex Exponential. I don't really understand it, nor do I understand its importance.
I know it is extremely important, and need to understand why, and what exactly it is, and the wording of the online resources...
Homework Statement
Calculate activation energy and pre exponential factor, i need help with the pre exponential factor
Homework Equations
Temperature °C
-20 = 253 k
-10= 263
0=273
10=283
20=293
Rate const. s-1
1.15
3.85
11.3
28.7
74.0
The Attempt at a Solution
i already plotted the graph on...
I recently completed a lab using an online projectile simulator about the range of projectiles. I launched a projectile with different initial speeds (5 m/s, 10 m/s, 15 m/s, 20 m/s, and 25 m/s). For each trial, I did the launch with and without air resistance and I plotted Range (with Air...
Homework Statement
Solve ##y=\mathrm{exp}(\frac{-x\pi}{\sqrt{1-x^2}})## for x when y = 0.1
Homework Equations
##\mathrm{ln}(e^x)=x##
The Attempt at a Solution
##\mathrm{ln}(0.1)=\frac{-x\pi}{\sqrt{1-x^2}}##
##(\frac{-\mathrm{ln}(0.1)}{\pi})^2=\frac{x^2}{1-x^2}##
Plz give me an easy explanation
I do know about the differentiation and second differentiation. I just don't get how that negetive sign comes in front of the exponent in the second differentiation
Hello, folks. I'm trying to figure out how to take the partial derivative of something with a complex exponential, like
\frac{\partial}{\partial x} e^{i(\alpha x + \beta t)}
But I'm not really sure how to do so. I get that since I'm taking the partial w.r.t. x, I can treat t as a constant term...
How do we get (6.265)?
Shouldn't we have
##exp(-i\frac{\alpha}{2}\hat{n}.\sigma)=\cos(\frac{\alpha}{2}\hat{n}.\sigma)-i\sin(\frac{\alpha}{2}\hat{n}.\sigma)##?
Homework Statement
https://dl.dropboxusercontent.com/u/17974596/Sk%C3%A6rmbillede%202016-02-02%20kl.%2007.35.26.png
I want to find variance matrix and expected variance vector of Y=(Y1,Y2). Y1 and Y2 are independant. Γ is the gamma function and ϒ is a known parameter. λ1>0 λ2>0 and ϒ>0...
Let's say my probability function is given by: p(y1,y2)=Γ(y1+y2+γ)/((y1+y2)!*Γ(γ)), when γ>0 is known. I suppose it is from an exponential family but I can't write in canonical form because I'm only familiar with exponential family with one variable so I'm confused now when there's to variable...
Homework Statement
Find the first and second derivative of the following function:
F(x)=e4ex
Homework Equations
d/dx ex = ex
d/dx ax = axln(a)
The Attempt at a Solution
I know the derivative of ex is just ex, but I'm not sure how to go about starting this one. I'm near certain I need to use...
This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{-x}dx=1.
Well, when we do Integration by parts we will have let u = x du = dx dv = e^{-x}dx v =...
Homework Statement
Hi, the problem is imply to show the following
\lim_{n\rightarrow \infty} 10^n e^{-t} \sinh{10^{-n}t} = \lim_{n\rightarrow \infty} 10^n e^{-t} \sin{10^{-n}t} = te^{-t}
How can I do this? Just a hint or a first step would be great, thanks :)
Homework EquationsThe Attempt...
Homework Statement
Solve the ODE: y''+x*y'-y=0
Homework EquationsThe Attempt at a Solution
Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2
In this case: y1=x, and I have the reached the integral below...