Logarithmic Action (2)!

Interact with the applet below for a few minutes. [br][b][color=#9900ff]At any time, you can move the purple point with coordinates ([i]c[/i], 0) anywhere on the positive x-axis. [/color] [/b][br][br]For any value [i]c[/i] > 0, can you write an identity that expresses the relationship visually shown here? [br][br]Algebraically show why this identity must be true for all values [i]c[/i] > 0.

Information: Logarithmic Action (2)!