### GRAPHS OF LOGARITHMIC FUNCTIONS

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

Vertical Shifts: A vertical shift takes place when a function is shifted up or down. The following examples look at the difference between the graph of f(x) and the graph of f(x) + C. For example, the graph of f(x) + 3 is the graph of f(x) shifted up 3 units. The graph of f(x) - 8 is the graph of f(x) shifted down 8 units.

Example 3: Graph the function and graph the function on the same rectangular coordinate system. Answer the following questions about each graph:

1. In what quadrants is 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 of the graph of the function ? What is the x-intercept and the y-intercept of the graph of the function ?
3. 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? What is the difference between the two points.
4. Describe the relationship between the two graphs.
5. How would you 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?
6. Describe the difference between the two equations.

1. You can see that the both graphs are located in quadrants I and II. Therefore, the domain (values of x) of both functions is the set of positive real numbers.
2. The graph of the function f(x) has an x-intercept at x = 1. The graph of the function g(x) has an x-intercept at x = 0.049787. You can see that neither of the graphs crosses the y-axis; therefore, neither of the graphs has a y-intercept.
3. The point , rounded to (2, 0.7) for graphing purposes, is located on the graph of .
4. The point , rounded to (2, 3.7) for graphing purposes, is located on the graph of . For each x-coordinate, the y-coordinates differ by 3.

5. Both graphs have the same shape. The graph of is nothing more than the graph of shifted up three units.
6. Shift (move) the graph of up 3 units. Every point on the graph of would be moved up 3 units. Therefore, the point (1, 0) would wind up at or after the move.
7. The only difference between the two equations is the + 3. Since , can be rewritten . This means that for every value of x, the function g(x) will always be 3 units larger than the function f(x).

Example 4: Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

1. In what quadrants is 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 of the graph of the function ? What is the x-intercept and the y-intercept of the graph of the function ?
3. 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?
4. Describe the relationship between the two graphs.
5. Describe how you would shift (move) the graph of so that it is superimposed on the graph of . Where would the point (1, 0) on the graph of be located after the move?
6. What is the difference between the two equations?

1. The graph of is located in quadrants I and IV, and the graph of is located in quadrants I and IV. This means that the domain (values of x) for both functions will always be positive real numbers.
2. The graph of crosses the x-axis at x = 1. Therefore, the x-intercept ix 1. The graph does not cross the y-axis; therefore, there is no y-intercept.
3. The graph of crosses the x-axis at 148.413159, rounded to 148.4 for graphing purposes:

Let and solve for x. Add 5 to both sides of the equation and we have .

Convert the equation to a logarithmic equation. .

The graph of g(x) does not cross the y-axis; therefore, the graph of g(x) has no y-intercept.

4. The point , rounded to (2, 0.7) for graphing purposes, is located on the graph of .
5. The point , rounded to (2, -4.3) for graphing purposes, is located on the graph of . For each x-coordinate, the y-coordinates differ by 5.

6. Both graphs have the same shape. The graph of is nothing more than the graph of shifted down 5 units.
7. Shift (move) the graph of down five units so that it is superimposed on the graph of . When we move the graph of down 5 units so that it is superimposed on the graph of , every point on the graph of is shifted down 5 units. Therefore, the point (1, 0) will be shifted down to or .
8. The difference between the two equations is the - 5. Since , can be rewritten .

If we knew that the point (c, d) is located on the graph of f(x), we can conclude that the point (c, d - 5) is located on the graph of g(x). For every value of x, the function g(x) will always be 5 units less than the function f(x).

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

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