What is Exponential function: Definition and 166 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.
I know that for 1 variable, one can write ##e^{f(x)} = \sum_{n = 0}^{\infty}\frac{(f(x))^n}{n!}##. In the case of 2-variables ##f(x,y)##, I assume we cannot write ##e^{f(x,y)} = \sum_{n = 0}^{\infty}\frac{(f(x,y))^n}{n!}## right (because of how the Taylor series is defined for multiple...
Hi PF
The logarithm is the inverse function to exponentiation. But, focusing on exponentiation, here comes the graph:
And, next, the quote
An exponential function is a function of the form ##f(x)=a^x##, where the base ##a## is a positive constant and the exponent ##x## is the variable...
I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...
Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing.
I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...
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!
Hi everybody
We can't differentiate ##x^x## neither like a power function nor an exponential function. But ##x^x=e^{x\mbox{ln}x}##. So
##\dfrac{d}{dx}x^x=\dfrac{d}{dx}e^{x\mbox{ln}x}=x^x(\mbox{ln}x+1)##
And here comes the doubt: prove the domain of ##x^x## is ##(0, +\infty)##
Why is only...
I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time.
The time-domain form is,
s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}}
Here, \mu , \Omega and \lambda are constants.
A quick...
Ok, first I tried to show that ##A = \left \{a^{r}|r\in\mathbb{Q},r<x \right \}## does not have a maximum value.
Assume ##\left\{ a^{r}\right\}## has a maximum, ##a^{r_m}##. By this hypothesis, ##r_{m}<x## and ##r_{m}>r\forall r\neq r_{m}\in\mathbb{Q}##. Consider now that ## q\in\mathbb{Q}|q>0##...
I try to proof it but i got stuck right here, i want your opinions
Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?
the integral is:
and according to mathematica, it should evaluate to be:
.
So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well:
however, I don't think this helps the issue much.
Homework Statement
-2^x = y
Homework EquationsThe Attempt at a Solution
When I plug this function in my graphing calculator, it appears to be 2^x reflected across the x axis.
This doesn't make sense to me. For example, for x values of 1 and 2, the value of y is not on the same half of the...
For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
Homework Statement
given the formula m=n*e^(-nt) show that the maximum of this curve is at m=1/(t*e^(1)).
2. The attempt at a solution
I can show this graphically but I am curious if it is possible to do it by hand?
I was wondering if anyone could point me to a set of rules for finding the base of an exponential function? I can figure out that the base of f(x)=7^x is 7 and the base of f(x)=3^{2x} is 9 but even though I know f(x)=8^{\frac{4}{3}x} has a base of 16, I don't know how that answer was reached.
How to Integrate it:::
∫e^(ax²+bx+c)dx
Or in general e raised to quadratic or any polynomial. I am trying hard to recall but I couldn't recall this integration. I tried using By-parts but the integration goes on and on.
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...
I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this?
P(r) = ∫p(r) dr = exp(-μr)
p(r) = dP(r)/(4πr²dr)
One way I tried to...
1. The problem statement, all variables, and given/known data
Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...
Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f).
I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?
Hey! :o
I am looking at the following:
Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$.
Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...
Homework Statement
Determine the Fourier-transfroms of the functions
\begin{equation*}
a) f : f(t) = H(t+3) - H(t-3) \text{ and } g : g(t) = \cos(5t) f(t)
\end{equation*}
and
\begin{equation*}
b) f : f(t) = e^{-2|t|} \text{ and } g : g(t) = \cos(3t) f(t)
\end{equation*}Homework Equations
The...
Hello all,
I have a complicated function:
\[f(x)=\left ( e^{x}+x \right )^{^{\frac{1}{x}}}\]
I need to find it's derivative and it's limit when x goes to infinity.
As for the derivative, I thought maybe to use LN, so that I can get rid of the exponent, am I correct?
How should I approach...
Homework Statement
The question is given just like this:
##\frac{d}{dx}(exp\int_1^x P(s)\ ds)## = ?
I assume they want me to find the derivative of the whole thing.
Homework EquationsThe Attempt at a Solution
I'm thinking the first step is:
##\frac{d}{dx}(exp\int_1^x P(s)\ ds) = (exp\int_1^x...
Homework Statement
a= eD/R*T*G make a linear equations
and calculate the value for D and G
R=8,3 and constant
D,G=constant
T= variable
Homework Equations
y=ax+b
y=numberax*bThe Attempt at a Solution
ax= E/(R*T)
x= 1/T
a= E/R
y= (E/R)*x+G
I don't know how to move on and if this is even correct/
Homework Statement
[/B]
Solve the equation
## e^{2x}+2=e^{3x-4}##Homework EquationsThe Attempt at a Solution
I know by using Newton-Raphson method the problem can be solved, i however tried solving it as follows
##e^{3x-4}-e^{2x}-2=0, e^{3x}-e^{2x}.e^{4}-2e^{4}=0, p^3-p^2.e^4-2e^4=0...
1. Homework Statement
Find derivative of
y=e^(cos(t)+lnt)
Homework EquationsThe Attempt at a Solution
So just using the chain rule:
y'=e^(cos(t)+lnt)*(-sin(t)+1/t)
The answer in the back of the book is
y'=e^(cos(t))*(1-tsin(t))
The wave function is an exponential function, if I plot the real part of it, I don't get a wave graph like sine or cosine function, Why the wave function is not represented by a trigonometric ratio instead.
Also, the wave function cannot be plotted since it is imaginary, why is it imaginary?
Thanks
Hey guys,
I need your help. I am not sure if this is the right part of the forum to ask this question.
So I started reading papers about the Lyapunov Exponent, but there is something I do not understand in the formula. Why ? It seems logical that we want because we want to get the Exponent...
So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials.
Here's the problem
A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of...
Mod note: Changed title from "Differential Euler's Number"
1. Homework Statement
Find the derivative.
f(t)=etsin2t
The Attempt at a Solution
f'(t)=etsin2t(sin2t)(cos2t)(2)
However the book seems to say that there should be an extra "t" in the solution. Some help?
Homework Statement
Hi, I have come across this equation in modelling exponential growth and decay. I am wondering if it is possible to solve it algebraically or not?
Homework Equations
8000-1.2031 * e^ (0.763x)=(0.5992×e^0.7895x)
The Attempt at a Solution
Brought all e^(ax) values to one...
Homework Statement
Find the first and second derivative of the following function:
F(x)=e4ex
Homework Equations
d/dx ex = ex
d/dx ax = axln(a)
The Attempt at a Solution
I know the derivative of ex is just ex, but I'm not sure how to go about starting this one. I'm near certain I need to use...
I am multiplying a lognormal distribution by an function to scale it larger. While I know that scaling a lognormal distribution by a constant multiplier yields a lognormal distribution, in this case the multiplier is not a constant. Instead, smaller values from the lognormal distribution are...
How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function?
Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?
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...
Homework Statement
superman has been disabled by a nearby amount of kryptonite, which decays exponentially. If Superman cannot regain his power until 90% og the kryptonite disintegrates, then how long will it be before he regain his powers?
Use r=-0.138629. Round to the nearest day.
a. 4days...
Homework Statement
[/B]
I need to differentiate the exponential function i = 12.5 (1-e^-t/CR) and I need to plot a table so that I can do a graph of i against t but I'm not sure how. (CR is the equivelant of Capacitance 20 Micro Fards and Resistance 300 Kilo Ohms)Homework Equations
[/B]
How do...
I'm modelling a variable output Y which has a value of 1 at x=0.
I've noticed that in the system I'm modelling, as x increases, y increases at an exponentially decreasing rate, up until a limit of around 1.53. I view this as changes in x causing the Y value to increase by a max of 53%.
The...
Demonstrate that ##|e^{z^2}| \le e^{|z|^2}##
We have at our disposal the theorem which states ##Re(z) \le |z|##. Here is my work:
##e^{|z|^2} \ge e^{(Re(z))^2} \iff## By the theorem stated above.
##e^{|z|^2} \ge e^x##
We note that ##y^2 \ge 0##, and that multiplying by ##-1## will give us...
Homework Statement
The graph goes through the points (-2, 13) and (0, 5) and has the horizontal asymptote y = 4.
f(−2) = ____ therefore:
____(B^____ ) = ____
b =
The Attempt at a Solution
f(−2) = 13 therefore:
1 (B^-2 ) = 13
b = ? not sure
Thank you
When I was first introduced to a derivation of the taylor series representation of the exponential function here (pg 25): http://paginas.fisica.uson.mx/horacio.munguia/Personal/Documentos/Libros/Euler%20The_Master%20of%20Us.pdf
I noted the author, Dunham mentioning that the argument was non...
So this question is from Khan Academy. I understood the first part and chose the correct function, but the second question(from 40 degrees to 30 degrees change) explanation confused me.
_____________________________________________________________________________
QUESTION:
Ajay made a...
This is part of a differential equations group project problem where I solve a set of differential equations to obtain the solution to a function. The part that I am stuck at involves taking the log of an exponential function, though there may be a mistake on the book's part, but I'm not sure...
Homework Statement
Prove that y(x,t)=De^{-(Bx-Ct)^{2}} obeys the wave equation
Homework Equations
The wave equation:
\frac{d^{2}y(x,t)}{dx^{2}}=\frac{1}{v^{2}}\frac{d^{2}y(x,t)}{dt^{2}}
The Attempt at a Solution
1: y(x,t)=De^{-u^{2}}; \frac{du}{dx}=B; \frac{du}{dt}=-C
2...
OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate e^{-2(x+y)} over a diamond that is centered around the origin:
\int\int_D e^{-2x-2y} dA
where D=\{ (x,y): |x|+|y| \leq 1 \}
I know that the region I'm integrating over is symmetric...
Hello everyone,
I was solving this problem:
Justify that ln6= ln2+ln3
So: exp(ln2+ln3)=exp(ln2)*exp(ln3)= 2*3= 6 = exp(ln6)
Till here, my work was okay.
What I didn't understand is : why should we say that the exponential function is strictly increasing over R before being able to simplify the...