### GRAPHS OF LOGARITHMIC FUNCTIONS

In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. Recall that whenever there is a minus sign in front of the logarithmic term, it means that there is a reflection across the x-axis. Recall also that whenever there is a minus sign in front of the x in the argument, it means that there is a reflection across the y-axis.

Horizontal and Vertical Shifts and Reflection Across the x-axis: The following examples discuss the difference between the graph of f(x) and the graph of -f(x + A) + B

Example 11: 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. 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 ?
3. Find the point (2, f(2)) on the graph of and find (4, g(4)) on the graph of . What do these two points have in common?
4. Describe the relationship between the two graphs.
5. 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 (1, 0) on be after the move?

1. You can see that the both graphs are located in quadrants I and IV.
2. You can see that neither of the graphs crosses the y-axis; therefore, neither of the graphs has a y-intercept. Notice that the graph of f(x) crosses the x-axis at 1 because . The graph of g(x) crosses the x-axis at 22.085537 because when x = 22,085537.
3. The point is located on the graph of . The point is located on the graph of .
4. Both graphs have the same shape. It appears that the graph of is the result of reflecting the graph over the x-axis, then shifting the graph to the right and upward.
5. After we reflect the graph across the x-axis, the point (1,0) stays the same. After the graph is then moved to the right 2 units and up 3 units, it would be superimposed on the graph of . The point (1, 0) on the graph of would first be shifted to the right 2 units and up 3 units to (1 + 2, 0 + 3) or (3, 3).

Example 12: Graph the function and the function on the same rectangular coordinate system. 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 (2, f(2)) on the graph of and find (6, g(6)) on the graph of
5. What do these two points have in common?
6. Describe the relationship between the two graphs.
7. Describe how you would move the graph of moved so that it would be superimposed on the graph of . Where would the point (1, 0) on the graph of wind up on after the move?

1. The graph of is located in quadrants I and IV. The graph of is located in quadrants II, III and IV.
2. The domain of is the set of all positive real numbers. The graph of is the set of real numbers less than 8.
3. The graph of does not cross the y-axis. It crosses the x-axis at 1.
4. The graph of crosses the x-axis at -12.085537 because when x = -12.085537. It crosses the y-axis at -0.920558 because .

5. The point is located on the graph of . The point is located on the graph of .
6. The point (6, g(6)) is located to the right and down from the point (2, f(2)).
7. Both graphs have the same shape. The graph of opens to the left whereas the graph of opens to the right. There are horizontal and vertical shifts.
8. Whenever there is a minus in front of the x, it means there is a reflection across the y-axis. Whenever something is added or subtracted from the x in the argument, it means there is a horizontal shift. Whenever a constant is added to the logarithmic term, it means there is a vertical shift.

9. Rewrite the equation by factoring out -1 from the argument to read .

From the equation, you can see that there is a reflection across the y-axis (minus sign in front of the x), a horizontal shift will be to the right 8 units, and a vertical shift down 3 units.

Therefore, reflect the graph of over the y-axis and then shift (move) the reflected graph to the right 8 units and down 3 units. The point (1, 0) would be located (-1, 0) after the reflection and (1 + 8, 0 - 3) or (9, - 3) after the horizontal and vertical shifts..

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

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