In mathematics, an exponential function is a function of the form
where b is a positive real number, and in which 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, as 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}.}
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:
For b = 1 the real exponential function is a constant and the derivative is zero because
log
e
b
=
0
,
{\displaystyle \log _{e}b=0,}
for positive a and b > 1 the real exponential functions are monotonically increasing (as depicted for b = e and b = 2), because the derivative is greater than zero for all arguments, and for b < 1 they are monotonically decreasing (as depicted for b = 1/2), because the derivative is less than zero for all arguments.
The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:
Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by
While both notations are common, the former notation is generally used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.
The exponential function satisfies the fundamental multiplicative identity
This identity extends to complex-valued exponents. 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
}
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>0.}
The fundamental multiplicative identity, along with the definition of the number e as e1, shows that
e
n
=
e
×
⋯
×
e
⏟
n
terms
{\displaystyle e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ terms}}}}
for positive integers n and relates 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 (for example, a matrix).
Its ubiquitous occurrence 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 (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences; thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
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 can be arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function (see above). 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.
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