Section 5.3
Logarithmic Functions
Objectives
1. Understanding the Definition of a Logarithmic Function
2. Evaluating Logarithmic Expressions
3. Understanding the Properties of Logarithms
4. Using the Common and Natural Logarithms
5. Understanding the Characteristics of a Logarithmic Function
6. Sketching the Graphs of Logarithmic Functions Using Transformations
7. Finding the Domain of Logarithmic Functions
Objective 1: Understanding the Definition of a Logarithmic Function
Every exponential function of the form f ( x )  b x where b  0 and b  1 is one-to-one and thus has an
inverse function. (You may want to refer to Section 3.6 to review one-to-one functions and inverse
functions). Remember, given the graph of a one-to-one function f, the graph of the inverse function is a
reflection about the line y  x . That is, for any point  a, b  that lies on the graph of f, the point  b, a  must lie
on the graph of f 1 . In other words, the graph of f 1 can be obtained by simply interchanging the x and y
coordinates of the ordered pairs of f ( x )  b x .
f ( x)  b x
The graph of f ( x)  b x , b  1
and its inverse.
f 1
(1, b )
(0,1)
( 1, b1 )
(b,1)
(1, 0)
( b1 , 1)
To find the equation of f 1 , we follow the 4-step process for finding inverse functions that was discussed in
Section 3.6.
Step 1. Change f ( x ) to y:
y  bx
Step 2. Interchange x and y:
Step 3. Solve for y:
x  by
??
Before we can solve for y we must introduce the following definition:
Definition of the Logarithmic Function
For x  0, b  0 and b  1 , the logarithmic function with base b is defined by
y  logb x if and only if x  b y .
The equation y  logb x is said to be in logarithmic form while the equation x  b y is in exponential form.
We can now continue to find the inverse of f ( x )  b x by completing step 3 and step 4.
Step 3. Solve for y:
x  b y can be written as y  logb x
Step 4. Change y to f 1 ( x ) :
f 1 ( x)  logb x
In general, if f ( x )  b x for b  0 and b  1 , then the inverse function is f 1 ( x)  logb x . For example, the
inverse of f ( x )  2 x is f 1 ( x)  log 2 x which is read as “the log base 2 of x.”
Objective 2: Evaluating Logarithmic Expressions
Because a logarithmic expression represents the exponent of an exponential equation, it is possible to evaluate
many logarithms by inspection or by creating the corresponding exponential equation. Remember, the
expression logb x is the exponent to which b must be raised to in order to get x.
For example, suppose we are to evaluate the expression log 4 64 . In order to evaluate this expression, we must
ask ourselves, “4 raised to what power is 64?” Since 43  64 , we conclude that log 4 64  3 .
For some logarithmic expressions, it is often convenient to create an exponential equation and use the method of
relating the bases for solving exponential equations. For more complex logarithmic expressions, additional
techniques are required.
Objective 3: Understanding the Properties of Logarithms
Since b1  b for any real number b , we can use the definition of the logarithmic function ( y  logb x if and only
if x  b y ) to rewrite this expression as logb b  1 . Similarly, because b0  1 for any real number b , we can
rewrite this expression as logb 1  0 . These two general properties are summarized below.
General Properties of Logarithms
For b  0 and b  1 ,
(1) logb b  1 and
(2) logb 1  0 .
In Section 3.6 we saw that a function f and its inverse function f 1 satisfy the following two composition
cancellation equations:


f f 1 ( x)  x for all x in the domain of f 1 and
f 1  f ( x)   x for all x in the domain of f
If f ( x )  b x , then f 1 ( x)  logb x . Applying the two composition cancellation equations we get


f f 1 ( x)  b
logb x
 x and
f 1  f ( x)   logb b x  x
Cancellation Properties of Exponentials and Logarithms
For b  0 and b  1 ,
log x
(1) b b  x and
(2) logb b x  x .
Objective 4: Using the Common and Natural Logarithms
There are two bases that are used more frequently than any other base. They are base 10 and base e. (Refer to
Section 5.2 to review the natural base e). Since our counting system is based on the number 10, the base 10
logarithm is known as the common logarithm.
Instead of using the notation log10 x to denote the common logarithm, it is usually abbreviated without the
subscript 10 as simply log x .
The base e logarithm is called the natural logarithm and is abbreviated as ln x instead of loge x .
Most scientific calculators are equipped with a log key and a ln key. We can apply the definition of the
logarithmic function for the base 10 and for the base e logarithm as follows:
Definition of the Common Logarithmic Function
For x  0, the common logarithmic function is defined by
y  log x if and only if x  10 y .
Definition of the Natural Logarithmic Function
For x  0, the natural logarithmic function is defined by
y  ln x if and only if x  e y .
Objective 5: Understanding the Characteristics of Logarithmic Functions
To sketch the graph of a logarithmic function of the form f ( x)  logb x where b  0 and b  1 , follow these 3
steps.
Step 1:
Start with the graph of the exponential function y  b x labeling several ordered pairs.
Step 2:
Since f ( x)  logb x is the inverse of y  b x , we can find several points on the graph of
f ( x)  logb x by reversing the coordinates of the ordered pairs of y  b x .
Step 3:
Plot the ordered pairs from step 2 and complete the graph of f ( x)  logb x by connecting the
ordered pairs with a smooth curve. The graph of f ( x)  logb x is a reflection of the graph of
y  b x about the line y  x .
Every logarithmic function of the form y  logb x where b  0 and b  1 has a vertical asymptote at the y-axis.
The graphs and the characteristics of logarithmic functions are outlined below
Characteristics of Logarithmic Functions
For b  0 , b  1 the logarithmic function with base b is defined by y  logb x .
The domain of f ( x)  logb x is  0,   and the range is  ,   . The graph of f ( x)  logb x has one of the
following two shapes.
f ( x)  logb x
( b , 1)
( b , 1)
f ( x)  logb x
f ( x)  logb x , b  1
f ( x)  logb x , 0  b  1
The graph intersects the x-axis at 1,0  .
The graph intersects the x-axis at 1,0  .
The graph contains the point  b,1 .
The graph contains the point  b,1 .
The graph is increasing on the interval  0,   .
The graph is decreasing on the interval  0,   .
The y-axis ( x  0 ) is a vertical asymptote.
The function is one-to-one.
The y-axis ( x  0 ) is a vertical asymptote.
The function is one-to-one.
Objective 6: Sketching the Graphs of Logarithmic Functions Using Transformations
Often we can use the transformation techniques that were discussed in Section 3.4 to sketch the graph of
logarithmic functions.
Objective 7: Finding the Domain of Logarithmic Functions
The domain of a logarithmic function consists of all values of x for which the argument of the logarithm is
greater than zero. In other words, if f ( x)  logb  g ( x) then the domain of f can be found by solving the
inequality g ( x)  0 .