Exponential function

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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