GRAPHS OF LOGARITHMIC FUNCTIONS

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

Combination of stretch, shrink, reflection, horizontal and vertical shifts. The following examples discuss the difference between the graph of f(x) and Af(B+Cx)+D. When A is negative, there is a reflection across the x-axis. When C is negative, there is a reflection across the y-axis. If B does not equal 0, there is a horizontal shift. If D does not equal zero, there is a vertical shift.

Example 17:

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 (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. 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 (e,1) on be after the move?

1. You can see that the graph of the function f(x) is located in quadrants I, II, and III The graph of the function g(x) is located in quadrant II.
2. 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 negative 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 -1.262452, rounded to -1.3 for graphing purposes. You could have determined the x-intercepts analytically without a graph. Note that Ln(x) = 0 when x = 1. Note that . This can be rewritten . This last equation can be converted to the exponential equation . When we solve for x in this last equation, we have = -1.262452. .
4. The point is located on the graph of f(x). The point is located on the graph of .
5. Even though the graph of g(x) looks difference from the graph of f(x), both graphs have essentially the same shape. The graph of g(x) results after you reflect the graph of f(x) over the x-axis, over the y-axis, shrink it, shift it to the right, and then shift it up.
6. The graph of f(x) is reflected over the x-axis, reflected over the y-axis, shifts to the right, is stretched by 3, and then shifted up.

Example 18:

Graph the function and the function on the same rectangular coordinate system.

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 (10, f(10)) on the graph of and find (10, g(10)) on the graph of .
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 (1000, 3) on the graph of wind up on after the move?

1. The graph of f(x) is located in quadrants I and IV The graph of g(x) is also 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 83.74466.

5. The point is located on the graph of . The point is located on the graph of .
6. Both graphs have essentially the same shape. The graph of g(x) is located below the graph of f(x).
7. When the graph of f(x) is shifted to the right , stretched by a factor of 5, and shifted down 11 units, it will be superimposed on the graph of g(x). The point (1000, 3) on the graph of f(x) would be shifted to (1000, ) = (1000,5.495356) on the graph of g(x).

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

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