### GRAPHS OF LOGARITHMIC FUNCTIONS

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

Horizontal Shifts: This is where the graph is shifted to the right or to the left on the rectangular coordinate axis. The following examples discuss the difference between the graph of f(x) and f(x + C). For example, the graph of f(x - 5) is the graph of f(x) shifted to the right 5 units. The graph of f(x + 3) is the graph of f(x) shifted to the left 3 units.

Example 5: Graph the function and the graph function on the same rectangular coordinate system. Answer the following questions about each graph:

1. State the domains of both functions.
2. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
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 (8, g(8)) on the graph of . What do these two points have in common?
5. Describe the relationship between the two graphs.
6. How would you shift (move) the graph of so that it is superimposed on the graph of ? After you move the graph, where would the point (1, 0) on be located?

1. The domain of the function f(x) is the set of positive real numbers. The domain of the function g(x) is the set of positive real numbers greater than 6.
2. You can see that the both graphs are located in quadrants I and IV. This means that the domains of both functions are subsets of the set of positive real numbers.
3. You can see that neither of the graphs cross 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 and the graph of g(x) crosses the x-axis at 7, 6 units to the right of 1.
4. The point , rounded to (2, 0.7) for graphing purposes, is located on the graph of .
5. The point , rounded to (8, 0.7) for graphing purposes.

6. Both graphs have the same shape. The graph of is nothing more than the graph of shifted to the right 6 units.
7. Shift (move) the graph of to the right 6 units. Every point on the graph of is thus moved to the right 6 units. The point (1, 0) is shifted to the right 6 units to or .

Example 6: Graph the function and the function on the same rectangular coordinate system. Answer the following questions about each graph:

1. What is the domain of each of the two function?
2. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
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. Describe how you would move the graph of so that it is superimposed on the graph of . Where would the point (1, 0) on the graph of wind up on after the move?
7. Describe the difference between the two equations.

1. The domain of the function f(x) is the set of positive real numbers. The domain of the function g(x) is the set of real numbers where x + 4 > 0 or the set of real numbers greater than - 4..
2. The graph of is located in quadrants I and IV. This verifies that the domain is the set of positive real numbers. The graph of is located in quadrants I, II and III to the right of - 4. This verifies that the domain of g(x) is the set of real numbers greater than - 4.
3. The graph of f(x) crosses the x-axis at 1 and never crosses the y-axis. The graph of g(x) crosses the x-axis at -3, 4 units to the left of 1. The graph of g(x) crosses the y-axis at Ln(4) or 1.386294.
4. The point , rounded to (2, 0.7) for graphing purposes, is located on the graph of . The point , rounded to (-2, 0.7) for graphing purposes, is located on the graph of .
5. Both graphs have the same shape. The graph of is nothing more than the graph of shifted to the left 4 units.
6. Shift (move) the graph of to the left 4 units. It will then be superimposed on the graph of . When you move the graph of to the left 4 units, every point on the graph of is shifted to the left 4 units. Therefore, the point (1, 0) will be shifted left to or .
7. The only difference in the two equations is in their arguments. The argument in the g(x) equation is 4 units greater than the argument in the f(x) equation.

Hint: Many students have a hard time determining whether the shift is to the left or to the right. One easy way to determine the answer is to set the argument equal to zero and solve. If the answer is negative, the shift is to the left. If the answer is positive, the shift is to the right.

For example, in the equation , the shift would be to the right because when x = 10. In the equation , the shift would be to the left because when x = -10.

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

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