WARM UP
1.
2.
Find the equation of the inverse relation
3
for xy  y3  2
xy  x  2
1
f
Find (x) for f(x) = 8x – 1.
Switch y and x in y = 8x – 1
x = 8y – 1
x + 1 = 8y
x 1
y
8
x 1
So f 
8
1
EXPONENTIAL &
LOGARITHMIC FUNCTIONS
OBJECTIVES

Graph exponential functions

Graph logarithmic functions

Model real-world problems that involve
exponential and logarithmic functions.
INTRODUCTION
 We have defined exponential notation for rational exponents. Let us
consider 2. The number π has an unending decimal representation.
3.1415926535…..
 Now consider this sequence of numbers.
3,
3.1,
3.14,
3.141
3.1415
3.14159…..
 Each of these numbers is an approximation to  . The more
decimal places, the better the approximation. Let us use these
rational numbers to form a sequence as follows:
3.14159
23
2 3.1
2 3.14
2 3.141
23.1415
2
 Each of the numbers in this sequence is already defined, the
exponent being rational. The numbers in this sequence get closer
and closer to some real number. We define that number to be 2 
 We can define exponential notation for any irrational exponent
x
in a similar way. Thus any exponential expressions a ,a  0, now
has meaning, whether the exponent is rational or irrational.
EXPONENTIAL FUNCTIONS
 Exponential functions are defined using exponential
notation.
Definition
x
The function f (x)  a , where a is some positive real-number
constant different from 1, is called the exponential function,
base a.
 Here are some exponential functions:
f(x)  2
x
h(x)  (0.178)x
1 x
g(x)  ( )
2
 Note that the variable is the exponent. The following are not
exponential functions:
f(x)  x
2
g(x)  x
1
3
h(x)  x 0.178
 Note that the variable is not the exponent.
EXAMPLE 1
Graph
y2 .
x
Use the graph to approximate
2
2
,
We find some solutions, plot them and
then draw the graph.
x
y
0
1
1
2
2
4
3
8
-1
½
2
¼
-3
⅛
Note that as x increases,
the function values
increase. Check this
on a calculator.
As x decreases, the
function values
decrease toward 0.
To approximate 2 2 we
locate 2 on the x-axis,
at about 1.4.
Then we find the corresponding
function value. It is about 2.7.
2 2  2.7
2
TRY THIS…
Graph y 
x
y
3x .
1
2
Use the graph to approximate 3 .
EXAMPLE 2
 We can make comparisons between functions using
transformations.
Graph
y4
x
We note that 4  (2 )  2x .
Compare this with y  2 graphed
in Example 1.
x
2 x
2x
Notice that the graph of y  2
approaches the y-axis more x
rapidly than the graph of y  2
2x
The graph of y  2 is a shrinking
of the graph of y  2 x
2x
Knowing this allows us to graph
y  22x at once. xEach point on
the graph of y  2 is moved half
the distance to the y-axis.
y  2 or4
2x
x
y2
x
TRY THIS…
Graph y  8 .
x
x
y
EXAMPLE 3
Graph
1x
y
2
We could plot some points and
connect them, but again let us
note that 1 x 1
x
( )  x 2
2
2
Compare this with the graph of
x in Example 1.
y2
x
The graph of y  2 is a reflection,
across the y-axis, of the graph of
y2
x
Knowing this allows us to graph
y  2x at once.
y2
x
or 1
2x
y2
x
TRY THIS…
1x
Graph y  ( ) .
3
x
y
LOGARITHMIC FUNCTIONS
Definition
A logarithmic function is the inverse of an exponential
function.
 One way to describe a logarithmic function is to interchange
variables in the equation y = a x . Thus the following equation
is logarithmic
y
xa
 For logarithmic functions we use the notation
which is read “log, base a, of x.”
y  log a x
means
xa
log a (x) or loga x
y
 Thus a logarithm is an exponent. That is, we use the symbol
y
to denote the second coordinate of a function.
xa
log a x
MORE LOGARITHMIC FUNCTIONS
The most useful and interesting
logarithmic functions are those for
which a > 1. The graph of such a
x
function is a reflection of y  a
across the line y = x. The domain
of a logarithmic function is the st of
all positive real numbers.
y  ax
ya
y
EXAMPLE 4
y  log 3 x
The equation y  log x is equivalent to x  3.y The graph of x  3y is a
3
Graph
reflection of y  3x across the line
y = x. We make a table of values
for y  3x and then interchange x
and y
For
y3
x:
x
y
For
0
1
1
3
y  log3 x
2
9
-1
-2
x
y
1
0
3
1
9
2
⅓
1/3
-1
1/9
1/9
-2
or y
x3:
Since a = 1 for any a ≠ 0, the graph
of y  log a x for a has the x-intercept
(1, 0)
y  3x
y  log3 x
TRY THIS…
y  log x
Graph
. What is the domain of this function?
2
What is the range?
x
y
CH. 12.1 & 12.2
Textbook pg. 519 #2, 6, 12 & 14
pg. 525 #2, 6, 14, 30 & 32