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.
Homework Statement
Suppose that the velocity v(t) (in m/s) of a sky diver falling near the Earth's surface is given by the following exponential function, where time is measured in seconds.
v(t) = 55 (1-e-0.18(t))
Find the initial velocity of the sky diver and the velocity after 6...
Homework Statement
Using the complex exponential, nd the most general function f such that
\frac{d^2f}{dt^2} = e-3t cos 2t , t all real numbers.
Homework Equations
I'm having a lot of trouble with this question, my thinking is to integrate once and then one more time...
I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question.
Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...
Good day!
I have a question regarding the law of the ff:
$$
\int_0^t h(s) e^{2\beta(\mu(s) + W_s)}
$$
where $\beta >0;$ $h,\mu$ are continuous functions on $\mathbb{R}_+$ with $h\geq 0;$
and $W=\{W_s,s\geq 0\}$ is a standard Brownian motion.
Thanks for any help.:D
Homework Statement
A town's population grows at 6.5% per annum. How many are in town now, if there will be 15 000 in 4.5 years?
Please explain which of these solutions is best. Or explain a better solution, please.
Homework Equations
A(t) = Per(t) <- general approach
A = P(1+i)t...
Homework Statement
So, i am given 3^x(2x) = 3^x + 2x + 1
And i want to solve for x.
Homework Equations
I only know that the solution is x=1 but i don't know how to get there.
The Attempt at a Solution
3^x(2x) = 3^x + 2x + 1
3^x(2x) - 3^x = 2x + 1
3^x(2x - 1) = 2x + 1
3^x =...
Homework Statement
I have the derived function:
f'(x) = [1/(1+kx)^2]e^[x/(1+kx)]
k is a positive constant
Homework Equations
I need to find the second derivative, which I thought was just the derivative of the exponent multiplied by the coefficient (as you find the...
I ask members here kindly for their assistance. I'm having some confusion over the process of integrating inequalities, in particular for obtaining the series expansion for the exponential function by integration. The text by Backhouse and Holdsworth (Pure Mathematics 2), shows the expansion of...
Homework Statement
Waiting time in a restaurant is exponentially distributed variable, with average of 4 minutes. What is the probability, that a student will in at least 4 out of 6 days get his meal in less than 3 minutes?
Homework Equations
The Attempt at a Solution
If I...
In page 11 of http://math.arizona.edu/~zakharov/BesselFunctions.pdf, I am trying to follow the derivation using binomial theorem to get this step:
(e^{j\theta}-e^{-j\theta})^{n+2k}≈\frac{(n+2K)!}{k!(n+k)!}(e^{j\theta})^{n+k}(-e^{-j\theta})^kIf you read the paragraph right above this equation...
I am trying to evaluate the following integral. Any help would be appreciated.
$$\int e^{x^2}dx$$
i tried the following,
$$x^2=t$$
$$2xdx=dt$$
$$\int\frac{e^t}{2\sqrt{t}}dt$$
i tried doing by parts but it didn't work
Hi,
Homework Statement
If the life expectancy of a light bulb is a random exponential variable and equal (on average) to 100 hrs, is λ then equal to 0.01 or to 100? (λ = 1/expectation)
Hello!
What is the integration of the absolute value of e^ix? That is what is ∫|e^ix|^2 equal to? The whole absolute thing got me lost. Thanks in advance.
I'm working with the integral from 0 to infinity of
t^(x-1)e^(-atcos(b))cos(atsin(b))
with respect to t. specifically, I'm asked to solve in terms of the gamma function. my question is more of what general technique i should use. all I've been able to do so far is beat it to death using...
A rabbit population satisfies the logistic equation dy/dt=2*10^-7y(10^6-y) where t is the time measured in months. The population is suddenly reduced to 40%of its steady state size by myxomatosis.
a) If the myxo' then has no effect how large is the population 8 months later?
b)How long will it...
Hello everyone, how are you?
I'm having trouble to evalue the following limit:
\lim_{x->\infty} (\frac{x}{1+x^2})^x
I "transformed" it into e^{ln{(\frac{x}{1+x^2})^x}} and tried to solve this limit:
\lim_{x->\infty} x ln{(\frac{x}{1+x^2})}
But I have no idea how to solve it correctly. Can...
Can someone explain how these are equivalent.
sqrt((-3)^2) = (-3)^2/2
=sqrt(9) and (-3)^1
3 is not equal to -3
(-3)^2/2 can be expressed as:
(-3^2)^1/2 and (-3^1/2)2
(9)^1/2 and (sqrt(-1)sqrt(3))^2...
the half life of C14 is 5730 years. if a sample of C14 has a mass of 20 micrograms at time t = 0, how much is left after 2000 years?
I learned from somewhere that these exponential decay and half life problems use the equation
y = ab^t or y = a(1+r)^t
where y = total, a = initial...
Homework Statement
[10] The air pressure in an automobile's spare tire was initially 3000 millibar.
Unfortunately, the tire had a slow leak. After 10 days the pressure in the tire had declined to 2800 millibar. If P(t) is the air pressure in the tire at time t,then P(t) satis es the di...
I'm looking for the expected value of an exponential Gaussian
Y=\text{exp}(jX) \text{ where } X\text{~}N(\mu,\sigma^2)
From wolframalpha, http://www.wolframalpha.com/input/?i=expected+value+of+exp%28j*x%29+where+x+is+gaussian
E[Y]=\text{exp}(j^2\sigma^2/2+j\mu)
If I were to use the...
Homework Statement
If f(x) = e^{3x^2+x} , find f'(2)Homework Equations
f'(x) = a^{g(x)}ln a g'(x)The Attempt at a Solution
f'(x) = (e^{3x^2+x})(ln e)(6x+1)
f'(2) = (e^{3(2)^2+2})(ln e)(6(2)+1)
= 2115812.288
I was checking online and I'm seeing a different answer, but this is EXACTLY how...
So in my math class we're studying derivatives involving ln(), tanh, coth, etc..
I need to say this first. I skipped precalc and trig and went straight to calculus, so whenever I see a trig problem, I can only go off of what I've learned "along the way." This problem has baffled me, please...
Hi,
I need to find out how to plot my data with exponential binning.
To better see the exponent of f(x) = x ^ \alpha, where x and f(x) are given, I am asked to do exponential binning the data.
Would appreciate you help.
Yours
Atilla
Any help is appreciated, thanks.
Homework Statement
In my course of differentials equations we were given the task to model a real life system with them, we choosed something that resembles a pendulum.Homework Equations
The Attempt at a Solution
We went to the lab and got experimental data from...
If I have ln(e^(-8.336/10c)) wouldn't that be the same as ln(e^(1/(8.336/10c))) therefore = 1/(8.336/10c) = 10c/8.336? I am confused about this because in my lecture notes they simplified ln(e^(-8.336/10c)) to just = -8.336/10c :confused:
Your help would be appreciated!
I wasn't quite sure where to put this, so here goes:
I am trying to find out some facts about the group SO(2,1). Specifically; Is the exponential map onto? If so, can the Haar measure be written in terms of the Lebesgue integral over a suitable subset of the Lie algebra? What is that subset...
Hi folks could someone please check my calculations contained in attached file?
thanks.
(incidentally, how can i create a link to such files in the future, weaving them into my text?)
Deus(has gone)
Problem: Suppose that $X \text{ ~ Exp}(\lambda)$ and denote its distribution function by $F$. What is the distribution of $Y=F(X)$?
My attempt: First off, I'm assuming this is asking for the CDF of $Y$. Sometimes it's not clear what terminology refers to the PDF or the CDF for me.
$P[Y \le y]=...
Hello!
Excuse me for my very basic understanding of math. I'll try and present my idea and problem clearly.
I'd like to devise a payout structure for a tournament.
20% of the entrants will be paid. The payout will be an exponentially sloping function. The payout is in percentages that equal...
Homework Statement
I have an adsorption reaction A+* \leftrightharpoons A^*
Then I am to finde the rate constants and these should be given by an Arhennius relation, e.g. from left to right in the reaction:
k_+ = f_+\exp(-\Delta E_a/k_bT) where the delta energy is the activation energy and...
Back to how income differences is explained in neoclassical theory, one can ask the question why GDP/capita is $5200 in Morocco and $35 500 in France. Real GDP per capita has grown at a rate of about 3.5% from 2008– 2012, GDP/capita in France grew in the same period at a rate of 1.1%...
Homework Statement
I want to know the steps involved in finding the magnitude of a complex exponential function. An example of the following is shown in this picture:
Homework Equations
|a+jb|=sqrt(a^2+b^2)
|x/y|=|x|/|y|
The Attempt at a Solution
For the denominator, I replaced z with e^jw...
[b]1. Homework Statement
I have a sequence whereby
10000=100(1+e^(kt)+e^(2kt)+...+e^(39kt)) where k=-4.7947012×10^(-3) which was dervied from dy/dt=ky
Re-arranging i get 99=1+e^(kt)+e^(2kt)+...+e^(39kt), letting e^(kt)=r I put it into the computer and
i get 1.04216=r=e^-4.7947012×10^(-3)t...
Hello all,
I'm having trouble showing that |e^it|=1, where i is the imaginary unit. I expanded this to |cos(t)+isin(t)| and then used the definition of the absolute value to square the inside and take the square root, but I keep getting stuck with √(cos(2t)+sin(2t)). Does anyone have any...
The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$
This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$
and vice versa...
Homework Statement
Given (e^(ix) - 1)^2 , show that it is equal to 2-2cosx
Homework Equations
e^ix = cosx + isinx
The Attempt at a Solution
After subbing in Euler identity and expanding I get:
cos(x)^2+sin(x)^2-2cosx-2jsinx+2jcosxsinx + 1
after using the addtion formulas I get...
Homework Statement
e^(i*2pi*1/15) is equal to ( e^(i*2pi) )^(1/15) = (1)^(1/15)=1
Why this is false?
Homework Equations
((A)^(b))^c=A^(b*c)=A^(bc). Why this isn't the case for complex exponential?
The Attempt at a Solution
Homework Statement
Reading Hinch's book, there is a statement as follows:
... z need to be kept in the sector where exp(-z^2) ->0 as z -> infinity. Thus it's applicable to the sector |arg z|<pi/4...Homework Equations
Why is this true and what is the limiting behavior of exp(x) for x in...
I have an EM problem (michelson interferometerish) where I have a term that I need to reduce. It is
|1+e^{ik \Delta cos\theta}|^{2}+| e^{ik \Delta sin\theta}|^{2}
I have foiled it and squared the last term but is there something that I am missing. I am multiplying it by a large matrix and...
Homework Statement
Show that the exponential distribution has the memory loss property.
Homework Equations
f_T(t) = \frac{1}{\beta}e^{-t/\beta}
The memory loss property exists if we can show that
P(X>s_1+s_2|X>s_1) = P(X>s_2)
Where...
Homework Statement
Given f(x;λ) = cx^{2}e^{-λx} for x ≥ 0
Determine what c must be (as a function of λ) then determine the maximum likelihood estimator of λ.
The Attempt at a Solution
So I'm supposed to integrate this from 0 to infinity, from what i can gather.
Let u = x^{2}, du...
Homework Statement
The total amount of oil in a well is 24000 barrels. The present rate of consumption is 100 barrels per year. How long will the gas supply last if the present yearly rate of consumption increases by 1% per year?
Homework Equations
We can use the approximation (1+x)^i =...
In class, my teacher was motivating Schrodinger's Equation, but there is one step that I do not understand or even have intuition for. I'll give the argument leading up to the step I do not understand for context.
Let \hat{U}(t) be the operator that gives the wave function after time t, given...
Homework Statement
This is a subset of a larger problem I'm working on, but once I get over this hang up I should be good to go. I have a set of measurements x_n that are exponentially distributed
p(x_n|t)=e^{-(x_n-t)} I_{[x_n \ge t]}
and I know that t is exponentially distributed as...