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
Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be...
Let X_1, X_2 \sim Exp(\mu) and Y = min(X_1, X_2) then find E[Y].
My attempt is as follows:
$$E[Y] = E[Y/X_1<X_2]P(X_1<X_2) + E[Y/X_1>X_2]P(X_1>X_2) \\
= \frac{1}{2} (E[X_1/X_1<X_2] + E[X_2/X_1>X_2] ) \\
= \frac{1}{2} (1/\mu + 1/\mu) \\
= 1/\mu$$
But, we know that minimum of two exponentially...
The formula f(t) = I(1 + r)t gives the value of something after it has grown for t years, at rate r, with an initial value I.
What if the growth rate r is continuously increasing?
What is the formula now?
Does it relate somehow to xx?
Hey, thanks for taking the time to look ay my post (:
I have attached a file which shows the question I am stuck on, and my attempt at working it out.
My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...
Our textbook defines an exponential function as
f(x) = ab^x. However, a question was brought up about a function, g(x) = 5^sqrt(x). Is g an exponential function? It looks like an exponential graph for x>0, but is not continuous on R.
Thanks in advance!
I'm trying to find a function which will assign exponential weights depending upon sample size.
nu=an equally space coefficient sequence (.05,..,.5; by=.05).
m=sample size (10 in this case)
Adding each observation in nu, implies a weight of 1, which makes the sum of weights m.
I need...
Homework Statement
Hi! I'm trying to find the PDF of W = abs(X-λ), where X is an exponential R.V. with rate parameter λ>0.
Homework Equations
The PDF for an exponential distribution is ∫λe^(-λx)dx.
Taking the derivative of a CDF will yield the PDF for that function (I'm aware there are...
Simplify 8p-3 x (4p2)-2/p-5.
This is what I got:
First thing I did was flip everything to get rid of the negative exponents:
p5/8p3x(4p2)2
next thing I did was multiply the 4p2 by 2
p5/ 8p3x16p4Then I subtracted thep5 from the 8p3
p2/8x16p4
and ended up with
1/128p2 I know that the...
Given that a random variable X follows an Exponential Distribution with paramater β, how would you prove the memoryless property?
That is, that P(X ≤ a + b|X > a) = P(X ≤ b)
The only step I can really think of doing is rewriting the left side as [P((X ≤ a + b) ^ (X > a))]/P(X > a). Where...
Hi
I have come across this equation:
zn-1=\bar{z}, z\inℂ* (ℂ*:=ℂ\0ℂ)
There are numerous obvious equalities that can be used, but I don't seem to reach a satisfying final answer.
Any help would be appriciated.
Thanks in advance :)
Easy exponential question---> 1/3(8)^(-2/3)
Homework Statement
I am doing a linear approximation question, and got stuck here with this simple stuff.
1/3(8)^(-2/3)
Can anyone explain what I need to do to turn the above into 1/12 ?
Homework Equations
The Attempt at a...
1. Homework Statement
- multiplication of trigonometric function and complex exponential
2. Homework Equations
the question is, Akcos(ωt) × e-jωt
3. The Attempt at a Solution
it is, Ak/2 + (Ak/2)e-j2ωt ?
by using cos(ωt) = 1/2ejωt + 1/2e-jωt
Homework Statement
(5 + √24)x + (5 - √24)x > 98
Homework Equations
inequalities
exponential
The Attempt at a Solution
I don't know how to start...trying to put log to both sides could not take me anywhere because the LHS can't be simplified...
How can we say:
f(x)=A'sin(kx)+B'cos(kx)
or equivalently
f(x)=Ae^{ikx}+Be^{-ikx}??
How are these two equivalent knowing that e^{ix}=cosx+isinx
I don't get this?
Hi,
Somebody asked me for some help with his calc II homework, and this was one of the questions:
"A hydra is a small freshwater animal and studies have shown that its probability of dying does not increase with the passage of time. The lack of influence of age on mortality rates for this...
Homework Statement
Hi,
I would like to get feedback if my z-plot is accurate for the following complex exponentiala:
a=2*exp(j*∏*t)
b=2*exp(j*∏*-1.25)
c=1*exp(j*∏*t)
d=-j*exp(j*∏*t)
Further analysis:
a= -2 because cos(∏) = -1 and sin(∏)=0
b= actual complex number A*cos(∅)+j*sin(∅)
c= -1...
Homework Statement
Determine the lowest derivative order for which the limit towards 0+ of the nth order derivative of f is nonzero (or otherwise does not exist). f = e^{\frac{-1}{x^{2}}}
Homework Equations
lim_{x\rightarrow0+}\frac{d^{n}}{dx^{n}}e^{\frac{-1}{x^{2}}}
The Attempt at...
Hi,
I have an equation with one unknown. This is the 'B' in the equation. I need to solve for 'B'. Can you help me to rearrange this complex equation to solve for B in terms of everything else please?
=(() - (DCRMgAl)) ∗ ^(−∗(()+()))+ ((DCRMgAl )- (DCRAl ))∗ ^(−∗∗() )+((Al) - (DCRMg)) ∗...
Homework Statement
Suppose that e^a=10 and e^b=4. Find the value of
loga(b)
Homework Equations
ln(b)=loga(b)/loga(e)
The Attempt at a Solution
So far I've only gotten to the above equation rearranged to:
ln(b) loga(e)=loga(b)/. I'm unsure as to where I'm supposed to go...
Homework Statement
When a camera flashes, the batteries begin recharging the flash capacitor which stores the charge Q according to the function Q(t) = Q* (1-e-t/a) where t is the elapsed time in seconds since the camera flash and Q* and a are non-zero
(a) What does Q* represent?
(b) Find the...
Homework Statement
Im not sure what each of the variables do in
y=cax-p +q
Homework Equations
The Attempt at a Solution
my understanding is that
-q is for vertical shifts/transformations
-p is for horizontal shifts/transformations
-and c is for vertical stretching
not...
I'm having trouble understanding the exponential map for nonlinear vector fields.
If dσ/dt=X(σ)
for vector field X, then how does one interpret the solution:
σ(t)=exp[tX]σ(0) ?
If X is nonlinear, then X is not a matrix, so this expression wouldn't make sense.
If X is a...
How do you divide using ln:
my example is x(7ln5-ln3)=-2ln5 and divide 7ln5-ln3 by both side to get x. So I have -2ln5/7ln5-ln3 and I can't seem to find the answer.
Homework Statement
dy/dθ =2·3^(−θ)
Homework Equations
The Attempt at a Solution
2*(d/dθ)(3^(-θ)
2(3^(-θ)ln(3))
(2ln(3))/(3^θ)
I keep getting this wrong and I'm not sure why. Could someone point out my mistake?
First-Order Linear System Transient Response
Hey there,
I am trying to solve a problem of first order equation which is
A temperature sensor has a first order response with τ = 18 seconds. The calibration curve of the sensor is presented in Figure 1. Graph the sensor response when it is...
Homework Statement
Hi,
This is probably an easy equation and I know this is the pre-cal section and this is for my engineering math class but I feel like people in a pre-cal forum should be able to help! I'm writing an essay for my engineering math class and I am trying to calculate the income...
Homework Statement
I just need some kind of explanation in layman's terms of what exactly is going on here. It seems as though I am missing some key element from trig. I am in a Signals class and the book lacks an explanation of the reduction used and ultimately why.
Homework...
Dear PF users
What is the rationale for Sze to write the exponential terms in ch. 1, Eq. 38 once as
\exp \left[ -\frac{E_C - E_F}{kT} \right] and once as \exp \left[ \frac{E_F - E_D}{kT} \right] ?
In both cases E_F is larger than either E_C or E_D. To me, for the sake of not having to worry...
I'm trying to understand the concept of exponential decay. To clarify, the decay of Uranium is not the same as say pulling the plug in a tub full of water. The water will drain out of a tub at a steady rate from start to finish. I assume Uranium doesn't do that. So you COULD say the water has a...
The derivative of e^(2x):
let y = e^(2x), let u = 2x, so y = e^u
chain rule: du/dx * dy/du = 2*e^u = 2e^u = 2e^(2x)
this is the solution copied from my book, my question is why do they let u = 2x? is e^u the same as e^x? If so then wouldn't all derivatives of the exponential functions be in...
Hi guys I'm helping my girlfriend with her Calc 1 homework and we seem to be stuck on this one problem. Am I missing something obvious?
Homework Statement
Homework Equations
P = P_0 a^t
\frac{5}{20} = \frac{P_0 a^1}{P_0 a^{-1}}
\frac{1}{4} = a^2
a =...
Hello forum,
I recently lost my notes on this matter, so I hope someone can fill in the gaps in my memory.
The problem I am working on is the following:
For one ticket window, the waiting time for one people satisfies an exponential distribution λe-λt and expected waiting time is 4 min...
WHY!? is the relationship between speed and time exponential?
I am actually a finance major but I like to learn about physics on my spare time. One of the more fundamental concepts I am having a hard time understanding is why the relationship between speed and time is exponential. Can someone...
Hi there,
I learn from the text that the exponential of an operator could be expanded with a series such that
e^{\hat{A}} = \sum_{n=0}^{\infty} \frac{\hat{A}^n}{n!}
So if the eigenvalue of the operator \hat{A} is given as a_i
e^{\hat{A}}|\psi\rangle
will be a matrix with diagonal elements...
Homework Statement
A radioactive substance diminishes at a rate proportional to the amount present (since all atoms have equal probability of disintegrating, ...). If A(t) is the amount at time t, this means that A'(t)= p * A(t) for some p representing the probability that an atom will...
I am reading the about the derivative of an exponential function using the limit definition, but one step I don't quite understand: lim_{h\rightarrow0}\frac{a^h -1}{h} = f'(0) Wouldn't that limit equal 0/0?
For an exponential function of the form y=a^x
First derivative , d/dx [a^x ] = a^x∙ lim┬(δx→0){(a^δx-a^0)/δx}
= a^x∙ m_((0,1))
now if m_((0,1)) which is the gradient of the y-axis intersection point of the exponential function equal to 1 exactly...
if first derivative is the slop of the given functions, then what is the physical meaning of exponential function remaining the same function after differentiation??
does it mean its vertical tangency make it indifferentiable?
please clarify me the concept...
regards
Homework Statement
(4x)^(1 + log(base 2) (x)) = 8(x^3)
What is the sum of the values of x that fullfill that equation?
A) 2.5
B) 2.0
C) 1.5
D) 1.0
E) 0.5Homework Equations
Use the exponential equation only and make the lower one (exponented) 1.The Attempt at a Solution
(4x) = (2 x^(1/2))^2...
Homework Statement
When attempting to divide the following equality (a "tangency condition" in microeconomic consumer theory), I'm puzzled by the solution derived here, please explain the procedure for arriving at the solution. Many thanks!
Homework Equations
I differentiated the...
Hello,
I'm having hard times with the following simple linear ODE coming from a control problem:
$$u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$$
with a given smooth α(t) satisfying
$$0 \leq \alpha(t) \leq u(t) \quad\mbox{for all } t\geq 0.$$
My intuition is that $$\lim_{t\to\infty}...
So I have a situation where the results in a table (x and y), where y reaches 0. According to my understanding, mathematically, exponential decay can never reach 0, right?
So does that mean that I can't use an exponential curve of best fit with the results in my table (x = 0,1,2,3), (y = 4...
You are given a random exponential variable X: f(x) = λ exp(-λ x).
Suppose that X = Y + Z, where Y is the integral part of X and Z is the fractional part of X:
Y = IP(X), Z = FP(X).
Which is the following conditional probability:
P(Z < z | Y = n) for 0 ≤ z < 1 and n = 0, 1, … ?
This may sound like a homework-type, but it isn't from any piece of homework, I just want to know how to do dimensional analysis by giving an example.
Let's have this simple equation:
k = a^{b}
Where k is a certain property of an object, a is the mass of the object, b is the length of...
Homework Statement
I'm trying to show that U(X+Y) = X in distribution, where X and Y are independent exp(λ) distributed and U is uniformly distributed on (0,1) independent of X+Y.Homework Equations
The Attempt at a Solution
X+Y is gamma(2,λ) distributed. But I can't figure out how to deal with...
Hey,
I'm currently reading a textbook which is attempting to derive the equation for a standing wave from first principles. I understand most steps with the exception of one.
It derives a sinusoidal function {x = A \sin \omega t} from a second order ODE, but then immediately interchanges...