# What is E^x: Definition and 129 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.

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27. ### Find Taylor series generated by e^x centered at 0.

1. a. Find Taylor series generated by ex2 centered at 0. b. Express ∫ex2dx as a Taylor series. 2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series: ex: 1 + x + x2/2! + x3/3! + ... ex2: 1 + x2 + x4/2! + x6/3! + ... For part b...
28. ### Symmetric arc length of ln(x) and e^x

Homework Statement Explain why ∫(1+(1/x2)1/2dx over [1,e] = ∫(1+e2x)1/2dx over [0,1] The Attempt at a Solution The two original functions are ln(x) and ex and are both symmetrical about the line y = x. If I take either of the functions and translate it over the line y = x the two...
29. ### Maximizing Planck's law using Taylor polynomial for e^x

Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...
30. ### Derivatives of e^x: Solutions and Explanations

Homework Statement Find the derivative of these e^x functions [paraphrased] Homework Equations x^{2} e^{x} The Attempt at a Solution 2x e^{x} This is how I believe it to be correct from my understanding. It does feel wrong, with the answers agreeing with the feeling.
31. ### Understanding the Limit of [1 + (1/z)]^z as z Approaches Infinity

Homework Statement Would you give me a clue as to how, limit as z approaches infinity, [[1 + (1/z)]^z]^(1/3) = e^(1/3)Homework Equations The Attempt at a Solution
32. ### Proove that e^x is always positive

May sound trivial but I don't find it trivial from the things I am given as a definition of the exponential function: The exponential function satisfies f(a+b)=f(a)f(b), is the inverse to the ln and is restricted by the condition: exp(x)≥1+x I can't see how I can from this only proove...
33. ### Express e^x from 1 to 8 as a Riemann Sum. Please, check my work?

Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :) 1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum. (Please ignore the __ behind the n's. The format is not kept without it...) _____n 2. lim Ʃ f(xi)(Δx)dx x→∞ i=1 Δx= (b-a)/n = 8-1/n = 7/n xi= 1 + 7i/n ____n lim Ʃ...
34. ### Evaluate lim e^x when x approaches zero from negative

Homework Statement What is the limit of e^x when x approaches zero from negative side Homework Equations The Attempt at a Solution Taylor series? Then the answer is put all x= 0 , and the answer is 1, but why the question ask from negative side?? Thank you very much
35. ### Modifying taylor series of e^x

I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course: \sum_{n=0}^\infty \frac{x^n}{n!} would change the equation. How would changing this to: \sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)} change the equation? Would it still be...
36. ### Inverse Function problem involving e^x

Homework Statement Let g(x) = (e^x - e^-x)/2. Find g^-1(x) and show (by manual computation) that g(g^-1(x)) = x. Homework Equations g(x) = (e^x - e^-x)/2 The Attempt at a Solution I get the inverse = ln[ (2x + sqrt(4x^2 + 4) ) / 2 ] How do I proceed?
37. ### Integrating e^x sin(lnx) using Integration by Parts

Homework Statement Integrate \int e2xsin(ln(x)) dx Homework Equations Well, I'm not exactly sure which rule to apply here, but I'm going to assume integration by parts: \int u \frac{dv}{dx} = uv - \int v \frac{du}{dx} The Attempt at a Solution I'm a little thrown off because...
38. ### Understanding the Integral of e^x: When is it Equal to xe^x?

Homework Statement I thought sometimes the integral of e^x is xe^x. Under what circumstances is the integral of e^x = xe^x? I think it has something to do with u substitution.
39. ### What are the rules for finding Maclaurin series for e^x?

Homework Statement I'm just trying to understand a few things about the Maclaurin series for e^x... So, in one case, if you have a series from 1 to infinity of [(-1)^n * 3^n ]/n!, how is it that it is equal to e^-3 - 1? I understand the e^-3 part, as -3 is simply our x value from the...
40. ### Proofing the derivatives of e^x from the limit approach

I was searching for the proof of \frac{d}{dx} e^x = e^x. and I found one in yahoo knowledge saying that \frac{d}{dx} e^x = \lim_{Δx\to 0} \frac {e^x(e^{Δx}-1)} {Δx} = \lim_{Δx\to 0} \frac {e^x [\lim_{n\to\infty} (1+ \frac{1}{n})^{n(Δx)}-1]} {Δx} Let h= \frac {1}{n} , So that n = \frac...
41. ### How can I integrate e^x arctan(x) without Wolfram?

I'm working on a take home exam in my Calculus 2 class. The exam is completely done except for one problem and I desperately need help. I've put so much time into this one problem that I'm ready to just miss it and take the hit. Homework Statement Indefinite Integral...
42. ### Fourier series (maybe) of e^x from 0 to 2pi

Hey, I have to show: Should I try to find the Fourier series from -2pi to 2pi? I have tried this already but I can't seem to get rid of the cos(nx/2) and sin(nx/2) to turn them into just sin(nx) and cos(nx) and the denominator stays as (n^2+4 instead of n^2+1. Any suggestions would be...
43. ### Which increases faster e^x or x^e ?

Homework Statement which increases faster e^x or x^e ?Homework Equations The Attempt at a Solution My attempt was taking the log of both, assuming it doesn't change anything (is this assumption correct?) x*ln(e) ------------------------ e*ln(x) now I took the derivative 1...
44. ### Derivative of e^x exponential functions.

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...
45. ### Indefinite Integral Question; e^x

Homework Statement Find the indefinite integral (preferably using u-substitution): ∫(ex-e-x)2 dx Homework Equations N/A The Attempt at a Solution To be honest, I'm slightly confused as to which path I'm supposed to take with this, especially since I'm not sure what I should be...
46. ### Solving a Limit: Evaluating limx->0 (e^x - 1- x - (x^2/2))/x^3

Homework Statement Evaluate limx->0 (e^x - 1- x - (x^2/2))/x^3 The Attempt at a Solution I can't remember how to solve this limit. Do I need to evaluate each part seperately? I plugged in the 0 to find that the limit does exist. I just can't seem to figure out what to do next.
47. ### A Why Question:Taylor Polynomial of e^x over x?

So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3? I hope my question makes sense... thanks
48. ### Infinite Solutions to e^x = c?

Hello, I have some background in complex analysis (a very minimal amount) but I did come up with a rather odd question. Given a polynomial a + bx + cx^2 + dx^3... nx^n There exists n or fewer solutions to the equation that each have a multiplicity of 1 to n. Given that information...
49. ### Integral of (ln(e^x + 1))^(1/3) / (e^x + 1)

Homework Statement Integral of (ln(e^x + 1))^(1/3) / (e^x + 1) See http://www2.wolframalpha.com/input/?i=integral+of+%28ln%28e**x+%2B+1%29%29**%281%2F3%29%2F%28e**x+%2B+1%29"Homework Equations N/A The Attempt at a Solution First, substitute k = ln(e^x + 1) dk = dx(e^x)/(e^x + 1) Then, used...
50. ### How to proove that that e^x is convex

Homework Statement I have to determine if [e][/x] is a convex function. If it is then show proof. I know its a convex function by looking at the graph, Iam stuck at prooving it mathematically though. Homework Equations The function is f(x)=e^x. The Attempt at a Solution I am...