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.
My trial :
I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##.
So any help about the right answer is appreciated!
From integration by parts, and using y(10) = 0, I get the equation ##2e^{3t-30} = \frac{|y-2|}{|y+1|}.##
Let ##f(t) = 2e^{3t-30}##.
Since it's for t>10, f(10) = 2, and we have ##2=\frac{|y-2|}{|y+1|}##. Depending on the sign I choose to use, I get either that y=-4 or y =0. Since ##t: 10...
Namaste
I seek a clarification on the periodicity condition of discrete-time (DT) signals.
As stated in Oppenheim’s Signals & Systems, for a DT signal, for example the complex exponential, to be periodic, i.e.
ej*w(n+N) = ej*w*n,
w/2*pi = m/N, where m/N must be a rational number.
Above is...
Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
Hello everybody!
I was studying the Glashow-Weinberg-Salam theory and I have found this relation:
$$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...
using this equation
##1-\sqrt{1-x^2/c^2}##
where c = 1 and x = 0.0 - 1.0 the speed of c
for example
##1-\sqrt{1-.886^2/1^2}## = y = 0.5363147619
gives me the y values. How do I find the inverse? How do find for x inputting the values of y?
Thank you.
Hi all:
I really do not know what to ask here, so please be patient as I get a little too "spiritual" (for want of a better word). (This could be a stupid question...)
I get this: eiθ=cosθ+isinθ
And it is beautiful.
I am struck by the fact that the trig functions manifest harmonic...
Hello,
I was wondering. Is the exponential function, the only function where ##y'=y##.
I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point.
Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...
Homework Statement
Homework Equations
The Attempt at a Solution
I understand there is no vertical asymptotes and can usually get the horizontal ,but can't understand with the exponential.
Homework Statement
I have the following data which I would like to model using an exponential function of the form y = A + Becx.
Using wolfram mathematica, solving for these coefficients was computed easily using the findfit function. I was tasked however to implement this using java and have...
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...
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?
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...
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...
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}##
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 Equations
The...
Homework Statement
The probability that a particular computer chip fails after a hours of operation is given by
0.00005∫e^(-0.00005t)dt on the interval [a, ∝]
i. find the probability that the computer chip fails after 15,000 hours of operation
ii. of the chips that are still operating after...
Hi,
I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit?
lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo
w and wo are both two points within the same plane.
what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number?
now i arrive at two different results by progressing in two different ways.
1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity.
2) now again if i...
Homework Statement
A bacteria culture initially contanis 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
a. Find an expression for the number of bacteria after t hours.
b. find the number of bacteria after 3 hours
c. find the rate of...
Homework Statement
Hi, I'm new here. I'm really rusty, I resume my career this year, and I'm reading 'the spivak book', (for Calculus 1).
Making some exercises, I get curious about how to solve this: x+e^x=4
I would love if someone could give me any trick
Homework Equations
The Attempt at a...
Homework Statement
The life times, Y in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.6 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if...
Homework Statement
In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...
Hi,
I need help to add an exponential trendline on my graph. I have three graphs and I have succeeded with the first one but somehow the exponential trendline will not show on the two other graphs even though I have selected the option multiple times. I also tried to re-do the graph in a new...