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.
View More On Wikipedia.org
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.
View More On Wikipedia.org
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- Tags
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Solving logarithmic and exponential equations
Homework Statement I'm trying to solve trying logarithmic and exponential equations but for the life of me I can't seem to figure out what I'm doing wrong. Homework Equations Log equation: log3(x+2)+1 = log3x²+4x Exp equation: 2x + 2x+1 = 3 The Attempt at a Solution For the first...- Mootjeuh
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- Exponential Logarithmic
- Replies: 8
- Forum: Precalculus Mathematics Homework Help
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Is j^{-p}=e^{-j\frac{p\pi}{2}} a valid exponential function?
Homework Statement I want to verify j^{-p}=e^{-j\frac{p\pi}{2}} Homework Equations e^{j\frac{\pi}{2}}=\cos(\frac{\pi}{2})+j\sin(\frac{\pi}{2})=j The Attempt at a Solution j^{-p}=(e^{j\frac{\pi}{2}})^{-p}=e^{-j\frac{p\pi}{2}} Am I correct? Thanks- yungman
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- Replies: 1
- Forum: Calculus and Beyond Homework Help
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MHB Transforming an integral of Exponential to a Contour
Hi friends, i need some help for this number: By considering the integral $$\int_{\gamma(0;1)}\exp(z) \mathrm{d}z$$,show that $$\int_0^{2\pi}\exp(\theta)\cos(\theta+\sin(\theta)) \mathrm{d}\theta = 0$$ i know that since $$f(z)=\exp(z)$$ is holomorphic on and inside...- shen07
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- Exponential Integral
- Replies: 4
- Forum: Topology and Analysis
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Solving Exponential Integral Homework
Homework Statement The problem says: u = 8[exp(y) - 1] Homework Equations The Attempt at a Solution Does this mean: 8ey-1 or 8ey-1- magma_saber
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- Exponential Integral
- Replies: 1
- Forum: Calculus and Beyond Homework Help
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MHB Find derivative with exponential function?
f(x)=x2ex the answer is f'(x)=(x2 + 2x)ex but I don't understand how to get there. Also I need to find g'(x) if g(x)=sqrtx(ex) would the answer for the second one be .5x-1/2ex?- coolbeans33
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- Replies: 5
- Forum: Calculus
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How Can You Solve 10^x = 2 Without a Calculator?
So, if we start with logarithm ln(2) assuming a base of 10, how would I solve this without a calculator? ln(2) 10^x = 2 And I get stuck there without a calculator. I could .. ln(10^x) = ln(2) x ln(10) = ln(2) x = ln(2) But that gets me back to where I started. So, how would I go...- Xaotique
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- Exponential Paper
- Replies: 4
- Forum: Set Theory, Logic, Probability, Statistics
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MHB Integration of exponential function
just want to confirm if i did set up my integral correctly and got a correct answer. $\displaystyle\int_0^a (e^{\frac{x}{a}}-e^{-\frac{x}{a}})$ using substitution for the first term in my integrand $\displaystyle u=\frac{x}{a}$ $\displaystyle du=\frac{1}{a}dx$; $\displaystyle dx=adu $ for...- paulmdrdo1
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- Replies: 1
- Forum: Calculus
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Real Solutions of Exponential Equations ##e^x = x^2## & ##x^3##
[1] Total no. of real solution of the equation ##e^x = x^2## [2] Total no. of real solution of the equation ##e^x = x^3## My Solution:: [1] Let ##f(x) = e^x## and ##g(x) = x^2## Now we have use Camparasion Test for derivative So ##f^{'}(x) = e^x## which is ##>0\forall x\in...- juantheron
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- Exponential
- Replies: 2
- Forum: Calculus and Beyond Homework Help
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Finding Electric Field of Exponential Charge Distribution
Homework Statement I am to find the electric field for a charge distribution of $$ \rho(x)= e^{-\kappa \sqrt{x^2}} $$ Homework Equations I know that gauss law is $$ \int E \cdot da = \frac{q_{enc}}{\epsilon_0} $$ The Attempt at a Solution I am not sure what the charge...- rodriguez1gv
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- Replies: 5
- Forum: Introductory Physics Homework Help
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Poisson distribution & exponential decay
Homework Statement t(s) = 1 15 30 45 60 75 90 105 120 135 N(counts) = 106 80 98 75 74 73 49 38 37 22 Consider a decaying radioactive source whose activity is measured at intervals of 15 seconds. the total counts during each period are given. What is...- cjurban
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- Replies: 11
- Forum: Introductory Physics Homework Help
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Exponential Decay function in NMR
Homework Statement How many oscillations occur before Mxy decays to approximately 1/3 of its initial value, for a Larmor frequency of 100 MHz and T2 of 100ms?Homework Equations I was learning about how NMR works and about transverse relaxation. According to what I learned, we can express...- Bubble22
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- Replies: 6
- Forum: Introductory Physics Homework Help
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Estimate exponential distribution parameter
Homework Statement A certain type of transistor has an exponentially distributed time of operation. After testing 400 transistors, it is observed that after one time unit, only 109 transistors are working. Estimate the expected time of operation. Homework Equations The Attempt...- Gauss M.D.
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- Replies: 1
- Forum: Calculus and Beyond Homework Help