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
lim_{x->0} (1+ sin5x)^{cotx}
Homework Equations
that's the problem.. I don't know :s
The Attempt at a Solution
can't think of any theorems or any methods I could use here.. what should I do? thank you
Homework Statement
Trying to solve for K:
ln(1-4k)=-6k
Homework Equations
The Attempt at a Solution
I know that need to take the e of both side, to get 1-4k=e^-6k, but I cannot remember any properties of e to allow me to remove the k from the exponent. Any tips is appreciated.
So I was playing around with logarithmic & exponential amplifiers in my lab class. I was looking at the following equations:
http://upload.wikimedia.org/math/7/7/6/77663157d5b97ceb2e3edac5f587a620.png and
http://upload.wikimedia.org/math/b/3/c/b3c569c85552561e41dec916f6e8ebe8.png...
Homework Statement
Hello i am doing a cooling experiment and i want to know the following:
y=28.49e^(-0.03890*800) +26.51 (note the 26.51) is not part of the exponential.
What is y? and I was wondering whether this would go ever go through the x axis?
Thanks people!
Homework...
Homework Statement
Find the second derivative of:
e^{ax}
and
e^{-ax}
Homework Equations
The Attempt at a Solution
The book that I am using seems to have been very vague on how to take the derivatives of exponential functions. I am aware that:
\frac {d(e^{x})}...
Exponential Equation! -Stuck!-
1. 32y+3 = 3y+5
The Attempt at a Solution
2y + 3 = y+5
2y = y+2
I really don't think I am doing this right, I am trying to find the value of Y. the answer is 2 but I don't know how to come up with the answer.
Hi,
I have the equation y'' +4y = t~sin(t)
i know that you usually guess the solution by substituting y for a polynomial (or whatever the form of the right side is).
But i want to do this by using the exponential function exp.
so, set y to equal te^{it}
chain rule:y^{\prime} = 1...
Homework Statement
We have the ODE y' = -ky + R for a population y(t) where death rate exceeds birth rate, counteracted by a constant restocking rate.
I'm assuming k is the decay constant and R is the restocking rate
The population at time t0 = 0 is y0, and I have to find a formula for...
Homework Statement
how would you integrate, int (-ik - 2ax)*exp(-2ax^2) dx with limits infinity - infinity
Homework Equations
i think i can use the result int exp(-x^2) dx = sqrt (pi). But I am stumped.
The Attempt at a Solution
i thought about multiplying it out
int...
Hello!
I found on this webpage:
http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/costate.pdf
page 1, on the bottom
that
e^{\phi^* a } f(a^{\dagger} , a ) = f(a^{\dagger} + \phi^*, a) e^{\phi^* a }
I have tried to prove this, writing both as taylor series, but the problem is to...
Homework Statement
Let z=x+iy prove that Exp[z1]*Exp[z2]=Exp[z1+z2]
Homework Equations
Binomial thm (x+y)^n=Sum[Bin[n,k]*x^n-k*y^k,{k,1,n}]
The Attempt at a Solution
I have no idea about this question...
Please give me some help.
I'm just a neuroscientist, so forgive me if the answer to this question is either obvious, or the answer is that it is impossible, obviously.
Basically, this image should outline the question clearly
http://img713.imageshack.us/img713/2569/mathsissuefixedyxs.gif
(And Y0 is always 1)Also, I...
I am reading conflicting interpretation of the Hubble constant in the exponentially expanding accelerating universe. Some say the Hubble constant is continuing to decrease; while others say Hubble constant is now unchanging and has become truly a constant.
Which is correct?
can someone please explain the difference. Graphically and mathematically it is easy to see they are inverses. But I see certain scales like the Richter scale that seem to increase exponentially, but are labeled as logarithmic scales. for example, on the richter scale with each increase in...
Homework Statement
I am working on a problem, and there is a small step I need help on:
I have the expression eab, where a is a constant, and b is a variable. I need to separate a and b so I can pull out the expression b from an integral expression. Is there any exponent law or manipulation...
Homework Statement
Homework Equations
The Attempt at a Solution
Ive numbered the solution steps, the ones that are giving me trouble are from 1 to 2 and from 3 to 4
From 1 to 2 i don't understand how there can be an exp(x) term taken out of the bracket and still be in the...
Homework Statement
Briefly, the question asks to prove how the interference of 2 electrons (travelling in opposite directions as 1-D waves) would affect the probability of finding each electron in free space. My issue has to do with the first step in the solution.
Homework Equations...
Homework Statement
problem 1
diff([f(tanx)], x) = x^2; prove that diff[f(u)]=(tan^-1(u))^2/(1+u^2)
problem 2
int(f(u), u = 0 .. 1) = f(t)-1; prove that f(a+b)=f(a)f(b) for every a,b ∈R
The Attempt at a Solution
problem 1
i don't know how to answer this...
I am learning calculus from a book of mine, and it gave an example problem of exponential growth (as derived from the exponential differential equation of dy/dx = ky to be y=Ce^kt) saying a population is growing at a rate of 2% per year, and says that K, the growth constant, in this case is k =...
Homework Statement
I don't understand why they took M(e)=1 , and how the proceed on with the proof.
Thanks in advance(=
Homework Equations
The Attempt at a Solution
Homework Statement
Hi all
Is infinity a minimum of the exponential function f(x)=ex or not? I personally do not think so since it is a limit, but I wanted to ask you guys to be 100% sure.
Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution.
e^(j*m*\theta1) + e^(j*m*\theta2)+e^(j*m*\theta3) + e^(j*m*\theta4) + e^(j*m*\theta5) = 0
where...
Homework Statement
\frac { d^2 \theta }{d x'^2 } = -y *exp(\theta) eq. 1
mayb be integrated to yield
exp(\theta) = \frac {a}{cosh^2(b \frac{+}{-} \sqrt \frac{a*y}{2} * x')}
\theta = f(y,x') Homework Equations
The Attempt at a Solution
the exponent is throwing me off, but i probably...
Let X1, X2, ...Xn be independent exponential variables having a common parameter gamma. Determine the distribution of min(X1,X2, ...Xn).
The Attempt at a Solution
I know how to do it with one X and one parameter but I am at a loss with these multiple ones...
Thanks so much!
Homework Statement
-Find the intervals on which f is increasing or decreasing
-Find the local maximum and minimum values of f
-Find the intervals of concavity and the inflection points
f(x)=xex
f'(x)= ex+xex
then I must solve for x when the function equals zero to find my critical numbers...
I have been given a Indice. I've been trying to figure it out for awhile and need some assistance, It'd be great if someone could work it out and show me the steps they did and even explain it.
([a-2b-3/ 2a3b-4]2 )/ [ab-15]/a-3b2])
I've attempted the working out in my book, Its abit hard...
Hello all,
I am trying to solve an integral with Mathematica, but I do not succeed. I am wondering whether the integral cannot be solved, or whether Mathematica cannot solve the integral, or whether I am doing something wrong.
Details:
* Mathematica does not seem to be able to solve the...
Homework Statement
y=9-x
Homework Equations
The Attempt at a Solution
dy/dx=-9-xln9
We are provided with the answers. I do not understand the concept behind how to get from the question to the solution. What rule(s) are applied? If somebody could solve this single question...
Homework Statement
i have to integrate e^(-y) / y
and i found out that you have to use this exponential integral and someone else said it doesn't have an integral. either way I am thoroughly confused
The Attempt at a Solution
i have no clue what so ever. The original question had it in...
Homework Statement
The bus service operates between points A and B. The buses depart from the terminal station A with headways that follow the negative exponentioal distributiona with parameter 10 trips per hour.
Assume that the characteristics of the system remain the same over time (not...
Hi all,
I'm trying to solve the definite integral between 0 and inf of:
exp(a*x^2 + b*x + c)
--------------------- dx
1 + exp(m*x + n)
with a,b,c,m,n real numbers and a < 0 (negative number so it converges).
I've read in the forum's rules that I have to post the work that I have...
Homework Statement
-
A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 63 cells.
-
(a) Find the relative growth rate.
Homework...
Homework Statement
y = x^(18x^-4)
Homework Equations
chain rule
dy/dx a^x = a^x ln a
The Attempt at a Solution
first i used the second equation from above to get
x^(18/x^4)*ln(x)
then i use the chain rule to get
x^(18/x^4)*ln(x)*(-72x^-5)
the computer program i am using is...
Solving Exponential Equations SO CONFUSED ?
Homework Statement
I'm trying to solve problems like these
a) 2(3y-2) = 18
b) 27(33x+1) = 3
c) 2x+2 - 2x = 48
d) 10z+4 + 10z+3 = 11
Homework Equations
Laws of exponential values. The Attempt at a Solution
b) 2(3y-2) = 18
6y-2 = 18...
Hi,
I have a (markov chain) transition matrix X which I understand. In particular each row of this matrix sums to 1.
I have used this transition matrix to construct it's generator, Y. I.e. Y is the continuously compounded transition matrix,
X = exp(Y)
X*X = exp(2Y), etc
both X and Y...
I'm trying to find the laplace transform (if possible) of exp(-2r/a)r^2.
---
I did integration by parts to check that the integral of exp(-2r/a)r^2 from 0 to infinity is a^3 / 4. but i cannot get the laplace transform to work out. the answer mathematica online gives would have to have the term...
Homework Statement
how can i convert this : - 2 (cos pai / 4 + i sin pai / 4 ) to Cartesian , Polar and Exponential form ?
Homework Equations
z = ( a + i b)
The Attempt at a Solution
r= -2
tan inverse = pai/4 / pai/4
??
Thank you very much for helping me out
Express -2(cos pai/4+i sin pai/4 ) in Cartesian , Polar and Exponential form ?
how can i convert this : - 2 (cos pai / 4 + i sin pai / 4 ) to Cartesian , Polar and Exponential form ?
Thank you very much
Homework Statement
Let <d1,d2,d3...dN> be an odered set of samples from an exponential
random variable with parameter lambda.
Let <l1,l2,l3,...,lM> the same.
Let Z = min<d1,d2,...,dN> --> Z is exp with parameter lambda*N
Let U = min<l1,l2,...,lM> --> U is exp with parameter lambda*M...
I have a problem with an exponential function. I am wondering if an exact solution is possible, or if I have to write the solution as a logarithm of an unknown.
A formula says that E(z)=E(0)^{-kz}, where E is light intensity and z is depth in water. My objective is to find the constant k. I...
I am having trouble solving this problem. I'm not sure how to solve this problem... Assume X and Y are independent exponential random variables with means 1/x and 1/y, respectively. If Z=min(X,Y): Is Z exponentially distributed as well (if so, how do you know)? What is the expectation of Z...
Homework Statement
for x and y satisfying x-y=2 and 2^{x}-2^{y}=6
find 2^{x}+2^{y}.
The Attempt at a Solution
x-y=2 ; x=2+y
log_{2} 2^{x}-log_{2}2^{y}=log_{2}6
log 6/log2=2.585
x-y=2.585 but this violates the intitial condition x-y=2
where am i going wrong?