Lesson 3-2
Logarithmic Functions
And
Their Graphs
Logarithmic Functions and
Their Graphs
When one looks at the graph of the exponential function
y = ax, by applying the horizontal line test, one can see
that this has an inverse function. This inverse function is
called a logarithm. The sole function of a logarithm is to
find an exponent!
Read the history of logarithms found at:
http://www.spiritus-temporis.com/logarithm/history.html
Logarithmic Functions and
Their Graphs
Definition of Logarithmic Function
For x > 0, a > 0, and a  1,
y = loga x
if and only if
x = a y.
The function given by
f(x) = loga x
is called the logarithmic function with base a.
Logarithmic Functions and
Their Graphs
Examples:
1. Evaluate f(x) = log4x when x = 64.
Solution: f(64) = log464 which means 4 to what power
equals 64?
43 = 64
Therefore, f(64) = log464 = 3 (Remember that
the purpose of a logarithm is to find an
exponent!)
Logarithmic Functions and
Their Graphs
Example 2:
Evaluate f(x) = log5x when x = 1/25.
1
1
 1 
f

log
which
means
5
to
what
power
gives
?
Solution:  
5
25
25
 25 
1
Remember that may be rewritten as 251 which
25
then leads to  5

2 1
or 52.
 1 
 1 
Therefore, f    log 5    2
 25 
 25 
Logarithmic Functions and
Their Graphs
Exampe 3:
Evaluate f(x) = log2x when x =1.
Solution:
f 1  log2 1 which means 2 to what power gives1?
Remember a  1.
0
Therefore, f 1  log2 1  0
Logarithmic Functions and
Their Graphs
Try the following:
Solutions:
1. f(x) = log6x when x = 216
f(216) = log6216 = 3
2. f(x) = log4x when x = 1
f(1) = log41 = 0
3. f(x) = log4x when x = 1/1024 f(1/1024) = log4(1/1024) = - 5
Logarithmic Functions and
Their Graphs
Some Basic Properties of Logarithms:
1. log a 1  0 because a  1.
0
2. log a a  1 because a  a.
1
3. log a a  x because a  a .
x
x
x
4. If log a x  log a y, then x  y.
Logarithmic Functions and
Their Graphs
Now let’s look at the graph of a logarithmic function.
Remember that when a graph has an inverse, the x- and
y-coordinates change positions. Look at the following
tables.
Inverse function
y = log3x
y = 3x
Its inverse is
x y
x
y
-3 1/27
1/27 -3
-2 1/9
1/9 -2
-1 1/3
1/3 -1
0
1
1
0
1
3
3
1
2
9
9
2
3 27
27
3
Logarithmic Functions and
Their Graphs
Graphing y = log3x
Logarithmic Functions and
Their Graphs
Now you graph y = log2x
Logarithmic Functions and
Their Graphs
Now let’s compare both graphs:
 0, 
 ,  
What is the domain for each graph?
What is the range for each graph?
What is the x-intercept for each graph?
1, 0 
What appears to be the vertical asymptote for each graph?
Is the function increasing or decreasing? If so, what is
increasing, the interval is  0, 
the interval?
x0
Logarithmic Functions and
Their Graphs
What would the function be with the following
transformations?
1. The reflection of y = logax?
y   log a x
2. A horizontal shift of 3 units to the left for y = log4x?
y  log4  x  3
3. A vertical shift of 5 units up for y = log6x?
y  log6 x  5
4. The reflection of y = logax with a horizontal shift of c
units and a verical shift of d units?
y   loga  x  c   d
Logarithmic Functions and
Their Graphs
The Natural Logarithmic Function
If y = ex is the natural exponential function,
then y = ln x is the natural logarithmic function.
For x > 0,
y = ln x if and only if x = ey.
The function is given by
f(x) = logex = ln x
is called the natural logarithmic function.
Logarithmic Functions and
Their Graphs
Graph the natural exponential function, the natural
logarithmic function, and the line of symmetry all on
one graph.
Logarithmic Functions and
Their Graphs
Properties of Natural Logarithms
1. ln 1 = 0
Why?
Because e 0 = 1
2. ln e = 1
Why?
Because e 1 = e
3. ln
ex
=x
Why?
4. If ln x = ln y, then x = y
Because in exponential form
if the two bases are the same,
then the exponents must be
equal.
Why?
Because since the two bases are the same, the two
results must be equal.
Logarithmic Functions and
Their Graphs
What you should know:
1. What a logarithm is
2. Basic properties for logarithms
3. How to graph a logarithmic function
4. Transformations to a logarithmic function
5. The natural logarithmic function and its basic properties