LOGARITHMS
Section 4.2
JMerrill, 2005
Revised 2008
Exponential Functions
 1. Graph the
exponential equation
f(x) = 2x on the graph
and record some
ordered pairs.
x
0
1
f(x)
1
2
2
3
4
8
Review
 2. Is this a function?
– Yes, it passes the
vertical line test (which
means that no x’s are
repeated)
 3. Domain?
 ,  
Range?
 0,  
Review
 2. Is the function oneto-one? Does it have
an inverse that is a
function?
– Yes, it passes the
horizontal line test.
Inverses
To graph an inverse, simply switch the x’s and
y’s (remember???)
x
f(x) =
f(x)
f -1(x) =
x
f(x)
1
0
0
1
1
2
2
1
2
4
4
2
3
8
8
3
Now graph
 f(x)
 f-1(x)
How are the Domain and Range of
f(x) and f -1(x) related?
The domain of the original function is the
same as the range of the new function and
vice versa.
f(x) =
x
f(x)
x
f(x)
0
1
1
0
1
2
2
1
2
4
4
2
3
8
8
3
f -1(x) =
Graphing Both on the Same Graph
Can you tell that the
functions are inverses
of each other? How?
Graphing Both on the Same Graph
Can you tell that the
functions are inverses
of each other? How?
They are symmetric
about the line y = x!
Logarithms and Exponentials
 The inverse function of the exponential
function with base b is called the logarithmic
function with base b.
Definition of the Logarithmic
Function
 For x > 0, and b > 0, b  1

y = logbx iff by = x
 The equation y = logbx and by = x are
different ways of expressing the same thing.
The first equation is the logarithmic form; the
second is the exponential form.
Location of Base and Exponent
Exponent
 Logarithmic: logbx = y
Exponent
Base
Exponential: by = x
The
1st
to the last = the middle
Base
Changing from Logarithmic to
Exponential Form
 a. log5 x = 2
means
52 = x
means
b3 = 64
 So, x = 25
 b. logb64 = 3
 So, b = 4 since 43 = 64
 You do:
 c. log216 = x
means
2x = 16
 So, x = 4 since 24 = 16
 d. log255 = x
means
25x = 5
 So, x = ½ since the square root of 25 = 5!
Changing from Exponential to
Logarithmic
 a. 122 = x
 b. b3 = 9
means
means
log12x = 2
logb9 = 3
 You do:
 c. c4 = 16
 d. 72 = x
means
means
logc16 = 4
log7x = 2
Properties of Logarithms
 Basic Logarithmic Properties Involving One:
 logbb = 1 because b1 = b.
 logb1 = 0 because b0 = 1
 Inverse Properties of Logarithms:
 logbbx = x because bx = bx
 blogbx = x because b raised to the log of some
number x (with the same base) equals that
number
Characteristics of Graphs
 The x-intercept is (1,0).
There is no y-intercept.
 The y-axis is a vertical
asymptote; x = 0.
 Given logb(x), If b > 1, the
function is increasing. If
0<b<1, the function is
decreasing.
 The graph is smooth and
continuous. There are no
sharp corners or gaps.
Transformations
Vertical Shift
 Vertical shifts
– Moves the same as all
other functions!
– Added or subtracted
from the whole function
at the end (or
beginning)
Transformations
Horizontal Shift
 Horizontal shifts
– Moves the same as all
other functions!
– Must be “hooked on” to
the x value!
Transformations
Reflections
 g(x)= - logbx
 Reflects about the x-axis
 g(x) = logb(-x)
 Reflects about the y-axis
Transformations
Vertical Stretching and Shrinking
 f(x)=c logbx
 Stretches the graph if
the c > 1
 Shrinks the graph if
0<c<1
Transformations
Horizontal Stretching and Shrinking
 f(x)=logb(cx)
 Shrinks the graph if the
c>1
 Stretches the graph if
0<c<1
Domain
 Because a logarithmic function reverses the
domain and range of the exponential
function, the domain of a logarithmic
function is the set of all positive real
numbers  0,   unless a horizontal shift is
involved.
Domain Con’t.
Domain
 0,  
Domain
 2,  
Domain
 4,  
Properties of Commons Logs
General
Properties
Common
Logarithms
(base 10)
logb1 = 0
log 1 = 0
logbb = 1
log 10 = 1
logbbx = x
log 10x = x
blogbx = x
10logx = x
Properties of Natural Logarithms
General
Properties
Natural
Logarithms
(base e)
logb1 = 0
ln 1 = 0
logbb = 1
ln e = 1
logbbx = x
ln ex = x
blogbx = x
elnx = x