5.4 Exponential Functions
Definition of the Natural
Exponential Function
 The inverse function of the natural logarithmic
function f(x)=ln x is called the natural exponential
function and is denoted by
 f-1(x)=ex
that is
 y=ex if and only if x = ln y
Inverse Relationship
 ln(ex) = x
And
 e ln x = x
Ex. 1 Solve the exponential equation
7 e
x 1
Ex. 2 Solve the exponential equation
ln( 2 x 3) 5
Operations with Exponential Functions
1. eaeb=ea+b
a
e
2.
b
e
e
a b
y=ex
Properties of the Natural Exponential Function
 The domain of f(x)=ex is (-∞, ∞) and the range is
(0, ∞)
 The domain of f(x)=ex is continuous, increasing and
one to one on its entire domain
 The graph of f(x)=ex is concave upward on its entire
domain
x
lim
e

x
0 and lim e x
x
The Derivative of the ex
 Let u be a differentiable function of x
1.
2.
d x
e
dx
e
d u
e
dx
du
e
dx
x
u
Ex 1 Find the derivative
d x
e
dx
1
Ex 2 Find the derivative
d x3
e
dx
Ex 3 Find the derivative
d
x
xe e
dx