First, let's study the effects of changing the a coefficient in f(x) = a · log_B(x + c) + d.

Follow the link below to launch an applet that will allow you to make changes to a.

(After opening the applet, use your mouse to click and drag the corner of the window to resize it if necessary. Close the browser tab/window to return to this lesson)

Log Function Applet

Notice you have an additional variable in the applet not mentioned in our equation, b, that could be changed. For our purposes we will leave b = 1 and ignore the effects it has on our equation.

• What is the original domain of the function before making any changes to a?
Interactive popup. Assistance may be required.
Check Your Answer
(0,∞)

You begin with the function f(x) = log₂x. Because a logarithm of a negative number (or zero) is undefined, the domain is restricted to positive numbers only.
• What is the range?
Interactive popup. Assistance may be required.
Check Your Answer
(-∞,∞).

Notice all y-values are covered by the graph.
• Does it have a vertical asymptote? If so, where?
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Check Your Answer
Yes, at x = 0. As x approaches zero from the right side, the graph gets very close to being vertical. We call this a vertical asymptote. The graph gets really close to x = 0 (the y-axis), but never actually touches it.
• Now, using the slider, change a to several different positive values, a > 0. Does this change your domain, range, or vertical asymptote?
Interactive popup. Assistance may be required.
Check Your Answer
No. Even though the graph grows slower and/or faster, the domain is still (0,∞), the range (-∞,∞), and asymptote x = 0.
• Next, slowly change a so that a < 0 until the shape of your graph reflects across the x-axis. Does this affect your domain, range, or vertical asymptote?
Interactive popup. Assistance may be required.
Check Your Answer
No. The domain is still (0,∞) and the range (-∞,∞), and asymptote x = 0.

Close the applet using the "Click here to close window" button.

Now, let's start over and study the effects of changing the parameter B (called the base) in our logarithmic form:

Log Function Applet

• Slowly decrease the value of B by moving the last slider to the left, but make sure B > 1. Then, increase B to the maximum value, 10.9. Do any of these changes, 1.1 < B < 10.9 cause a change in range?
Interactive popup. Assistance may be required.
Check Your Answer
No. The range remains (-∞,∞).
• Is your domain affected?
Interactive popup. Assistance may be required.
Check Your Answer
No. It is still (0,∞).
• Where is your vertical asymptote?
Interactive popup. Assistance may be required.
Check Your Answer
It is still located at x = 0.
• What does this tell you about the significance of a and B in exponential equations of the form
  f(x) = a · log_B(x + c) + d?
Interactive popup. Assistance may be required.
Check Your Answer
Although a can reflect the graph across the x-axis depending if it is positive or negative, it doesn't alter the domain, range, or asymptote. Also, as long as B > 0, there is no change in domain, range, or asymptote.

Close the applet using the "Click here to close window" button.