### 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 f(Cx).

Example 15: When C > 1.

Graph the function and the function on the same rectangular coordinate system. and 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 both functions.
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 (e, f(e)) on the graph of and find (e, g(e)) on the graph of . What do these two points have in common?
5. Describe the relationship between the two graphs.
6. Write g(x) in terms of f(x).
7. 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 (10, 6.234411) 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. The graph of f(x) crosses the x-axis at 1, and the graph of g(x) crosses the x-axis at . You could have determined the x-intercepts analytically without a graph. Note that Ln(x) = 0 when x = 1. Note that Ln(5x) = 0 when 5 x = 1 or x = 1/5.
4. The point is located on the graph of . The point is located on the graph of . This is a stretch.
5. Even though the graph of g(x) looks difference from the graph of f(x), both graphs have the essentially the same shape. The graph of g(x) is located above the graph of f(x) for all positive values of x, and the graph of g(x) is located below the graph of f(x) for all negative values of x.
6. The graph of is a result of stretching and shrinking the graph of . For example, for every positive value of x the value of g(x) is larger than the value of f(x) by Ln(5).

7. =
8. The graph of f(x) is stretched upward by adding Ln(5) to every y-coordinate. The point (10, 6.234411) on the graph of would wind up at (10, 6.234411+Ln(5)) after the move. If (a,b) is a point on the graph of f(x), then (a,b+Ln(5)) will be a point on the graph of g(x).

Example 16: When 0 < C < 1.

Graph the function and the function on the same rectangular coordinate system. Note that . This means that for each x coordinate, the y-coordinates will differ by Log(4).

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 both functions.
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 (100, f(100)) on the graph of and find (100, g(100)) on the graph of . What do these two points have in common?
5. Describe the relationship between the two graphs.
6. Write g(x) in terms of f(x).
7. Describe how you would move the graph of moved so that it would be superimposed on the graph of . Where would the point (1000, 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 of the graphs cross the y-axis. Therefore neither of the graphs has a y-intercept.
4. The graph of f(x) crosses the x-axis at 1, and the graph of g(x) crosses the x-axis at 4. when .

5. The point is located on the graph of . The point is located on the graph of . Note that 1.397940 = 2 - Log(4)
6. Both graphs have the same shape. The graph of g(x) is located below the graph of f(x).
7. = This means that the y coordinate of every point of the graph of g(x) is smaller by Log(4).
8. The point (1000, 3) on the graph of f(x) would be shifted to (1000,3-Log(4)) = (1000,2.39794) on the graph of g(x).

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

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