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.
In this text, I will ask a question about the power series expansion of exponential and logarithmic integrals.
Now, to avoid confusion, I will first give the definitions of the two:
\mathrm{Ei}(x)=\int_{-\infty}^{x}\frac{e^t}{t}dt
\mathrm{Li}(x)=\int_{0}^{x}\frac{dt}{\log(t)}
where Ei denotes...
Hello,
I need to find the CDF of
\mathcal{X}=\sum_{l=0}^L|h(l)|^2
where
h(l)
is complex Gaussian with zeros mean and variance
\sigma^2_l
In particular, I need to proof that:
\text{Pr}\left[\mathcal{X}\leq b\right]\doteq b^{L+1}
where dotted equal means in asymptotic...
Homework Statement
here's a problem from my assignment
let integral p(x)dx=Ae^-(x/a) dx...(1)
find value of A, that makes integral p(x)dx=1;
and
find mean x so that integral x*p(x)dx=a...2)
Homework Equations
now
to solve the first one i found out A to be (-1/a*e^(x/a))
but...
Homework Statement
Find the eigenfunctions (with angular momentum 0) and the estimation of the 3 first energy levels (given g and a) of a particle in a exponential potential such as
V = -ge-r/a
Homework Equations
Time independent Schrodinger equation (SE)The Attempt at a Solution
Did a...
Homework Statement
Show if true:
\sum_{i=1}^{n-1}e^{2 \pi ift}=\frac{e^{2 \pi ifn}-1}{e^{2 \pi if}-1}=e^{\pi if(n-1)}\frac{e^{\pi ifn}-e^{-\pi i f n}}{e^{ \pi if}-e^{-\pi if}}\\\\\\\\
Homework Equations
I'm really stuck here, just looking for a suggestion as to what equation to use...
Can I...
Hello gurus,
I've been trying to prove the following inequality for several days:
\int_1^\infty \frac{\exp\left(-\frac{(x-1)^2}{2a^2}\right)}{x}dx > \ln(1+a)\quad \forall a>0.
I've demonstrated by simulations that this inequality holds. I‘ve also proved that this inequality holds for large...
Homework Statement
In the exercise 8.4 from Quarks and Leptons. An Introductory Course in Modern Particle Physics - F.Halzem,A.Martin we can see:
if the charge distribution \rho(r) has an exponential form e^{-mr}, then:
F(q) \propto (1 - \frac{q^2}{m^2})^{-2}
where...
I'm having some trouble solving for t in the following exponential equation.
$$ B = A_1 e^{-\lambda_1 t} + A_2 e^{-\lambda_2 t} $$
I can't divide out the leading coefficients A1 and A2 because they differ. I'm not really sure how to immediately take the natural logarithm of both sides...
Homework Statement
Find the inverse of exp(y-x)+5
2. The attempt at a solution
I think the solution is y-ln(x-5) but I can't think of how to solve it to get that. I don't know what to do about the x and the y being together.
I'm reviewing for my final exam so one of the practice problems is:
5= 3^(x+5)
Here's my attempt at it:
ln 5= x+5 ln 3
ln 5 / ln 3 = x+5
(ln 5 / ln 3)-5= x
1.46-5 ≈ x
-3.54 ≈ x
I checked my answer and I get 3^1.46 ≈ 4.97 so rounding it up gives me 5 since I rounded off 1.46...
What exactly is the difference between the mean waiting time and the median waiting time for an exponential distribution? I'm looking for a slightly intuitive understanding. I know the formulae, with the mean waiting time as 1/λ and the median as ln2/λ (which I notice is also the formula for...
How would one evaluate $$\Phi = \int_{-\infty}^{+\infty} e^{-(ax+bx^2)} dx$$.
I was trying to change it into a product of an error function and a gamma function, but I needed an extra dx. Any other ideas?
Hi,
I am working on a modeling exercise and I would like to know which is the correct general expression for a Exponential Function:
y = abc(x+d)+e
or
y = abcx+d+e
Thanks in advance,
Peter G.
Homework Statement
Consider the complex number z=(i^201+i^8)/(i^3(1+i)^2).
(a) Show that z can be expressed in the Cartesian form 1/2+(1/2)i.
(b) Find the modulus of 4z − 2z*. (z* meaning z-bar/complex conjugate of z)
(c) Write 2z in polar form.
(d) Write 8z^3 in polar exponential form...
I need to compute the matrix exponential of a real matrix that has no special structure (is not symmetric, positive definite, nilpotent, normal, etc...). Currently, this is the most time consuming step in my program; I need to be able to transform this matrix, I suppose, into something that can...
Can anyone check my work? I'm doubtful of my answer.
Homework Statement
The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container?
Homework Equations
I used:
F = A × 260
(1/2)F = A × 2x
F...
I need help calculating the exponential map of a general vector.
Definition of the exponential map
For a Lie group G with Lie algebra \mathfrak{g}, and a vector X \in \mathfrak{g} \equiv T_eG, let \hat{X} be the corresponding left-invariant vector field. Then let \gamma_X(t) be the maximal...
Homework Statement
Find the MLE of θ = P (X≤ 2) in a random sample of size n selected from an exponential distribution EXP(λ)
Homework Equations
f(x, λ) = λ e^(-λx)
F(x, λ) = 1 - e^(-λx)
The Attempt at a Solution
I know how to find the MLE of the mean of an exponential...
I'm a very conceptual person, and I've been reading about exponential and logarithmic growth but don't fully have the kind of conceptual grasp on the two and how they differ that I'd like, so I"m curious:
What is the difference between exponential growth and logarithmic growth and what causes...
Hi there, I'm having a hard time trying to plot a Bode Diagram here in MatLab.
The transfer function not only has an exponential function in it, but also has 3 unknowns to which I am not supposed to assign value, and on top of that it is a complex number.
Let me write it down to you...
I would like to find derivations of exp(-ik0r) respect to k in order to calculate exp(-ik1r) by using Taylor expansion:
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This expansion converges when the value of r is relative low (0.3 - 1.2)...
I've been asked to prove that the following distribution is a member of the generalised exponential family of distributions.
f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2)
I know that i have to transform the equation into the form
f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)}
and that to do this i...
Homework Statement
I try to solve this integral with with parameter x as a member of this scale:(-∞ , +∞)
I=∫∏dx[i] exp(-0.5XAX + XB)=∫∏dx[i] exp( Ʃ-0.5x[i]a[i][j]x[j] +Ʃ x[i]b[i] )
In which a[i][j] and b[i] are components of telated matrix and vector and the first sum is on i and j ranges...
Homework Statement
A discrete random variable Y has probability distribution given by
f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2)Homework Equations
I know that for a pdf to be from generalised exponential family of distribution it can expressed as
f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)}The Attempt at...
Homework Statement
A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea?Homework Equations
Density...
Homework Statement
If we have the exponential distribution f_X(x)=\frac{1}{2}e^{-x/2} then show that the cumulative distribution function of Y=\sqrt{X} is given by F_Y(y)=1-e^{-y^2/2}Homework Equations
F_Y(y)=f_X(x)\cdot\left| \frac{dx}{dy}\right|
F_Y(y)=f_X(h^{-1}(y))\cdot\left|...
I'm taking the Fourier transform of a signal. This integral has bounds from -∞ to ∞, but since the signal is 0 for negative t, the bounds become 0 to ∞
doing the integration, the antiderivative I get is et*(-3-jω+2j) where j is sqrt(-1)
Now I have to evaluate this at t=infinity (since it is a...
I was just thinking, shouldn't hooke's law be represented with an exponential equation?
I am thinking that in class, the springs we use for the hooke's law experiment are small, but if they were very large, then the higher coils would be sustaining much more weight than the lower coils, and...
Homework Statement I am of the conclusion that, under any circumstance, the extended exponential rule can not be applied to (1+x)^{1/x}.
Thus, there is no way for the extended exponential rule to arise when taking the derivative of:
f(x) = (1+x)^{\frac{1}{x}}e^{x}
For instance, if my first...
Which of the following numbers can be added directly in exponential notation?
1. 3x10^-4
2. 600x10
3. 32.6x10^-3
4. 0.4x10^-4
_______________________________
a. 1 and 2
c. 1 and 4
d. 2 and 4
b. 1and3
Homework Statement
The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra.
Homework Equations
See above.
The Attempt at a Solution
I have been looking around, reading the textbook and...
Homework Statement
This question was asked in the Indian National Maths Olympiad.
The question is:
Find all the possible real ordered pairs of (x,y) for equation
16^[(x^2) + y] + 16^[x + (y^2)]=1
Homework Equations
That was the only equation.
The Attempt at a Solution...
Homework Statement
In the last couple threads, it has become apparent that I need to organize my understanding of some of the derivative rules, specifically as they relate to exponential functions, such as e^x.
So I wrote out a couple possible ways of evaluating e^x. Could you tell...
now i have a function defined on Z+
that is, it is defined on all positive integers, and it is complex
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how can i decompose the original function into the exponential functions? i.e...
Homework Statement
What is the derivative of y=x^(13/x^2) with respect to x?
The Attempt at a Solution
I went through multiple techniques to solve this, but all of them have failed so far ._.
In my latest attempt, I took the natural log of both sides:
lny= lnx^(13/x^2)
I...
Is that any way to find a finite value which is not equal to zero using L'hopital's rule in
limit(z=-ia)
exp[-A/(z+ia)]/(z+ia)^2
i kept getting 0/0 after differentiation
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Sum of reciprocal of some base (I just chose e as example) to prime power?
Ʃ \frac{1}{e^{p}} = \frac{1}{e^2}+\frac{1}{e^3}+\frac{1}{e^5}+\frac{1}{e^7}+\frac{1}{e^{11}}+\frac{1}{e^{13}}+\frac{1}{e^{17}}+...
p\inP
Brute force simulation gives me
~0.19279118970439518
Is there an...
Hi guys I have a problem in finding my error in a calculation, I will be glad if you help me to find the error that I am doing
ok the problem is basically about matrix exponentials, here we go:
A, B, U, P are matrices
n is a natural number
t and T are rational numbers and T=n*t
now in...
Homework Statement
Compute the derivative of the following function
(1-2x)e^{-x}
Homework Equations
Product Rule and Quotient rule
The Attempt at a Solution
My problem here is that I come up with two different answers when I use the quotient rule vs. the product rule.
Trying it with the...
Homework Statement
A biomedical company finds that a certain bacteria used for crop insect control will grow exponentially at the rate of 12% per hour. Starting with 1000 bacteria, how many will they have after 10h?
Homework Equations
Topic is application of differential equations...
1. Problem Statment:
Sketch the sequence x(n)=\delta(n) + exp(j\theta)\delta(n-1) + exp(j2\theta)\delta(n-2) + ...
3. Attempt at the Solution:
The angle theta is given in this case Can someone remind me of how to multiply a complex exponential by a delta function? This sequence represents...
Homework Statement
http://img4.imageshack.us/img4/224/32665300.png
The Attempt at a Solution
http://img684.imageshack.us/img684/2920/scan0003xo.jpg
I've uploaded my work so far since its much faster than typing and I'm stuck on the last line trying to solve the integral.
The first...
I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n!
and so
eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n!
I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...
Homework Statement
Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem.
Find b, such that maximum of the magnitude of the frequency response function...