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 where and 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. That is, for any point that lies on the graph of f, the point must lie on the graph of . In other words, the graph of can be obtained by simply interchanging the x and y coordinates of the ordered pairs of .
The graph of
and its inverse.
To find the equation of , we follow the 4-step process for finding inverse functions that was discussed in Section 3.6.
Step 1. Change to y:
Step 2. Interchange x and y:
Step 3. Solve for y: ??
Before we can solve for y we must introduce the following definition:
Definition of the Logarithmic Function
For , the logarithmic function with base b is defined by
if and only if .
The equation is said to be in logarithmic form while the equation is in exponential form. We can now continue to find the inverse of by completing step 3 and step 4.
Step 3. Solve for y: can be written as
Step 4. Change y to :
In general, if for , then the inverse function is . For example, the inverse of is 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 is the exponent to which b must be raised to in order to get x.
For example, suppose we are to evaluate the expression. In order to evaluate this expression, we must ask ourselves, “4 raised to what power is 64?” Since , we conclude that .
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 for any real number, we can use the definition of the logarithmic function ( if and only if ) to rewrite this expression as . Similarly, because for any real number, we can rewrite this expression as . These two general properties are summarized below.
General Properties of Logarithms
For ,
(1) and
(2) .
In Section 3.6 we saw that a function f and its inverse function satisfy the following two composition cancellation equations:
for all in the domain of and
for all in the domain of
If , then . Applying the two composition cancellation equations we get
and
Cancellation Properties of Exponentials and Logarithms
For ,
(1) and
(2) .
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 to denote the common logarithm, it is usually abbreviated without the subscript 10 as simply.
The base e logarithm is called the natural logarithm and is abbreviated as instead of .
Most scientific calculators are equipped with a key and a 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 the common logarithmic function is defined by
if and only if .
Definition of the Natural Logarithmic Function
For the natural logarithmic function is defined by
if and only if .
Objective 5: Understanding the Characteristics of Logarithmic Functions
To sketch the graph of a logarithmic function of the form where, follow these 3 steps.
Step 1: Start with the graph of the exponential function labeling several ordered pairs.
Step 2: Since is the inverse of , we can find several points on the graph of by reversing the coordinates of the ordered pairs of .
Step 3: Plot the ordered pairs from step 2 and complete the graph of by connecting the ordered pairs with a smooth curve. The graph of is a reflection of the graph of about the line.
Every logarithmic function of the form wherehas a vertical asymptote at the y-axis. The graphs and the characteristics of logarithmic functions are outlined below
Characteristics of Logarithmic Functions
For , the logarithmic function with base b is defined by .
The domain of is and the range is . The graph of has one of the
following two shapes.
, ,
The graph intersects the x-axis at . The graph intersects the x-axis at .
The graph contains the point . The graph contains the point .
The graph is increasing on the interval . The graph is decreasing on the interval .
The y-axis () is a vertical asymptote. The y-axis () is a vertical asymptote.
The function is one-to-one. 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 then the domain of f can be found by solving the inequality.