Transcript Sec 4.1 Exponential Functions

Sec 4.1 Exponential Functions
Objectives:
•To define exponential functions.
•To understand how to graph
exponential functions.
Exponential Functions
The exponential function with base a is defined
for all real numbers x by
f ( x)  a
where a > 0 and a  1.
x
Ex 1. Let f ( x)  3 and evaluate the
following.
x
a) f (2)
 2
b) f   
 3
c) f ( )
d) f
 2
Graphs of Exponential Functions
Ex 2. Graph each function.
x
1


x
a) f 3
b) f
 
 
3
HW p336 1-4,11,13
Sec 4.2 Logarithmic Functions
Objectives:
• To define logarithmic functions.
• To understand properties of log
functions.
Definition of the Logarithmic Function
Let a be a positive number with a  1. The
logarithmic function with base a, denoted by
loga, is defined by
log a x  y  a  x
y
Ex 1. Express the equation in
exponential form.
a) log101000 = 3
b) log232 = 5
c) log100.1= –1
d) log164 = ½
Ex 2. Express the equation in
logarithmic form.
a) 23  8
1
2
b) 81  9
c) 4

3
2
 0.125
d) 73  343
Properties of Logarithms
Property
1
loga1 = 0
2
logaa = 1
3
logaax = x
4
alogax = x
Reason
Ex 3. Evaluate the expression.
a) log 4 4
5
b) log 4 64
c) log 9 1
Ex 4. Use the definition of the
logarithmic function to find x.
a) log5 x  4
b) log 4 2  x
c) log x 25  2
The Natural Exponential Function
The natural exponential function is the
exponential function
f ( x)  e
with base e.
x
The Natural Logarithmic Function
The logarithm with base e is called
the natural logarithm.
ln x = logex
Properties of Natural Logarithms
Property
Reason
We must raise e to the power 0
to get 1.
1
ln 1 = 0
2
ln e = 1
We must raise e to the power 1
to get e.
3
ln ex = x
We must raise e to the power x
to get ex.
4
eln x = x
We must raise e to the power ln x
to get x.
Ex 5. Evaluate the expression.
a)
b)
e
ln 
ln e
4
1

c) ln  
e
d) eln
2
HW p349 3-35 odd