GRAPHS OF LOGARITHMIC FUNCTIONS

In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. We will also illustrate how you can use graphs to HELP you solve logarithmic problems.

(Reflection across the x-axis):

Whenever the minus sign (-) is in front of the function notation, it indicates a reflection across the x-axis. For example, the graph of - f(x) is a reflection of the graph of f(x) across the x-axis. The graph of 3 - g(x) involves the reflection of the graph of g(x) across the x-axis and the upward shift of the reflected graph 3 units.

Example 1:

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

  1. What is the domain of both functions?
  2. In what quadrants is the graph of the function located?
  3. In what quadrants is the graph of the function located?

  4. What is the x-intercept and the y-intercept on the graph of the function ?
  5. What is the x-intercept and the y-intercept on the graph of the function ?

  6. Find the point (2, f(2)) on the graph of and find ( 2, g( 2)) on the graph of .
  7. What do these two points have in common?
  8. Describe the relationship between the two graphs.
  9. How would you move the graph of so that it would be superimposed on the graph of ? Where would the point (1, 0) on be located after such a move?

The graph to the right of the y-axis is the graph of the function , and the graph on the left to the left of the y-axis is the graph of the function .

  1. The domain of both functions is the set of positive real numbers.
  2. You can see that the graphs of both functions are located in quadrants I and IV to the right of the y-axis. This verifies that the domain of both functions is the set of positive real numbers.
  3. The graphs of both functions cross the x-axis at x = 1. Since neither of the graphs cross the y-axis, there is no y-intercept.
  4. The point is located on the graph of . The point is located on the graph of .
  5. Note that both points have the same x-coordinate and the y-coordinate’s differ by a minus sign.
  6. Mentally fold the coordinate system at the x-axis. Note that the graph of the function is superimposed on the graph of the function .
  7. This means both graphs are symmetric to each other with respect to the x-axis. What exactly does that mean? Well for one thing, it means if there is a point (a, b) on the graph of , we know that the point (a, - b) is located on the graph of .

    The shapes are the same. The graph of is a reflection over the x-axis of the graph of .

  8. Fold the graph of over the x-axis so that it would be superimposed on the graph of . Every point on the graph of would be shifted up or down twice it’s distance from the x-axis. For example, the point (a, 8) is located 8 units up from the x-axis. If we shifted the point (a, 8) down 16 units, it would wind up at (a, - 8) units. If the point (b, -11) is located on the graph of , it would be shifted up 22 units to (b, 11). Since the point (1, 0) is on the x-axis, the point would not move.

If you would like to review examples on the following, click on Example:

Reflection over the y-axis: The graph of f(x) versus the graph of f(-x). Example.

Vertical shifts: The graph of f(x) versus the graph of f(x) + C. Example.

Horizontal shifts: The graph of f(x) versus the graph of f(x + C) Example.

Combination horizontal shift and reflection across the y-axis: The graph of f(x) versus the graph of f(- x + C) or f(C - x) Example.

Combination horizontal and vertical shifts: The graph of f(x) versus the graph of f(x + A) + B Example.

Combination horizontal and vertical shifts and reflections: The graph of f(x) versus the graphs of -f(x) + C. f(-x) + C, f(x + C) + D. Example.

Stretch and Shrink: The graph of f(x) versus the graph of C(x). Example.

Stretch and Shrink: The graph of f(x) versus the graph of f(Cx). Example.

Combination of stretch, shrink, reflection, horizontal, and vertical shifts: Example.

Solving an equation from a graph: Example.

[Exponential Rules] [Logarithms]

[Algebra] [Trigonometry ] [Complex Variables]

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Author: Nancy Marcus

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