### GRAPHS OF LOGARITHMIC FUNCTIONS

In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions.

Stretch and Shrink: The following examples discuss the difference between the graph of f(x) and the graph of Cf(x):

Example 13:

Graph the function and the function on the same rectangular coordinate system. Recall that you will have to convert the above functions to either the natural log or log with base 10 to graph them on your calculator. The function can be written as and the function can be written as for graphing purposes. Answer the following questions about each graph:

1. In what quadrants in the graph of the function located? In what quadrants is the graph of the function located?
2. State the domain of each function.
3. What is the x-intercept and the y-intercept on the graph of the function ? What is the x-intercept and the y-intercept on the graph of the function ?
4. Find the point (2, f(2)) on the graph of and find (2, g(2)) on the graph of . What do these two points have in common?
5. Describe the relationship between the two graphs.
6. How would you moved the graph of so that it would be superimposed on the graph of ? When you moved the graph, where would the point (9, 2) on be after the move?

1. You can see that the both graphs are located in quadrants I and IV.
2. The domain of both functions is the set of positive real numbers.
3. You can see that neither of the graphs crosses the y-axis; therefore, neither of the graphs has a y-intercept. Notice that both graphs cross the x-axis at 1 because when x = 1 and when x = 1. The y-coordinate of g(x) is 4 times the y-coordinate of f(x).
4. The point is located on the graph of . The point is located on the graph of .
5. Both graphs have the basic shape shared by all logarithmic functions. It appears that the graph of is a result of stretching the graph of . For example, for every value of x the value of g(x) is 4 times larger than the value of f(x).
6. The graph f(x) is stretched up to be superimposed on the graph of g(x). The point (9, 2) on the graph of would be moved up to (9,8) because every function value of g(x) is 4 times the function value of f(x). If (a, b) were a point on the graph of f(x), then the point (a, 4b) would be a point on the graph of g(x).

Example 14:

Graph the function and the function on the same rectangular coordinate system. Recall that you will have to convert the above functions to either the natural log or log with base 10 to graph them on your calculator. The function can be written as and the function can be written as for graphing purposes. Answer the following questions about each graph:

1. In what quadrants in the graph of the function located? In what quadrants is the graph of the function located?
2. State the domain of each function.
3. What is the x-intercept and the y-intercept on the graph of the function ? What is the x-intercept and the y-intercept on the graph of the function ?
4. Find the point (49, f(49)) on the graph of and find (49, g(49)) on the graph of . What do these two points have in common?
5. Describe the relationship between the two graphs.
6. Describe how you would move the graph of moved so that it would be superimposed on the graph of . Where would the point (343, 3) on the graph of wind up on after the move?

1. Both graphs are located in quadrants I and IV.
2. The domain of both functions is the set of positive real numbers.
3. Neither graph crosses the y-axis; therefore, there are no y-intercepts. The graph of crosses the x-axis at 1, and the graph of crosses the x-axis at 1.
4. The point is located on the graph of . The point is located on the graph of . For each x-coordinate, the y-coordinate of g(x) is of the y-coordinate of f(x).
5. Both graphs have the same shape. The graph of appears to shrink. It is like you just let the graph of f(x) kind of slip down a little to be superimposed upon the graph of g(x)..
6. Notice that the graph of g(x) is below the graph of f(x). This is because for every value of x, the value of g(x) is the value of f(x). The point (343,3) on f(x) would

be shifted to on g(x)

If you would like to review another example, click on Example.

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